lieonn.hh

/* You can use one of the both BSD 3-Clause License or GNU Lesser General Public License 3.0 for this source. */
/*
BSD 3-Clause License

Copyright (c) 2013-2025, bitsofcotton (kazunobu watatsu)
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright notice,
  this list of conditions and the following disclaimer in the documentation
  and/or other materials provided with the distribution.

3. Neither the name of the copyright holder nor the names of its
  contributors may be used to endorse or promote products derived from
  this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

*/

#if !defined(_SIMPLELIN_)

using std::max;
using std::min;
using std::abs;
using std::sqrt;
using std::exp;
using std::log;
using std::isfinite;

using std::move;
using std::swap;
using std::pair;
using std::make_pair;
using std::vector;
using std::map;

using std::stringstream;
using std::istringstream;
using std::ifstream;
using std::ofstream;
using std::istream;
using std::ostream;

using std::string;
using std::to_string;
using std::cerr;
using std::endl;
using std::flush;
using std::getline;

using std::sort;
using std::binary_search;
using std::lower_bound;
using std::unique;

// N.B. external linkage.
extern vector<string> words;
template <typename T, typename U> class corpus;


// N.B. start ifloat
// Double int to new int class.
template <typename T, int bits> class DUInt {
public:
  inline DUInt(const int& src = 0) {
    assert(0 < bits && ! (bits & 3));
    const auto abssrc(src < 0 ? - src : src);
    e[0]   = T(abssrc);
    e[1]  ^= e[1];
    if(abssrc != src)
      *this = - *this;
  }
  inline DUInt(const T& src) {
    const auto abssrc(src < T(int(0)) ? - src : src);
    e[0]   = abssrc;
    e[1]  ^= e[1];
    if(abssrc != src)
      *this = - *this;
  }
  inline DUInt(const DUInt<T,bits>& src) { *this = src; }
  inline DUInt(const DUInt<DUInt<T,bits>,bits*2>& src) { *this = src; }
  inline DUInt(DUInt<T,bits>&& src) { *this = src; }
  inline ~DUInt() { ; }
  
  inline DUInt<T,bits>& operator ++ () {
    ++ e[0];
    if(!e[0]) ++ e[1];
    return *this;
  }
  inline DUInt<T,bits>  operator ++ (int32_t) {
    const auto work(*this);
    ++ *this;
    return work;
  }
  inline DUInt<T,bits>& operator -- () {
    if(!e[0]) -- e[1];
    -- e[0];
    return *this;
  }
  inline DUInt<T,bits>  operator -- (int32_t) {
    const auto work(*this);
    -- *this;
    return work;
  }
  inline DUInt<T,bits>  operator -  () const {
    auto work(~ *this);
    return ++ work;
  }
  inline DUInt<T,bits>  operator +  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work += src;
  }
  inline DUInt<T,bits>& operator += (const DUInt<T,bits>& src) {
    // N.B. assembler can boost dramatically this code. but not here.
    const auto e0(max(e[0], src.e[0]));
    e[0] += src.e[0];
    if(e[0] < e0)
      e[1] ++;
    e[1] += src.e[1];
    return *this;
  }
  inline DUInt<T,bits>  operator -  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work -= src;
  }
  inline DUInt<T,bits>& operator -= (const DUInt<T,bits>& src) {
    return *this += - src;
  }
  inline DUInt<T,bits>  operator *  (const DUInt<T,bits>& src) const {
    DUInt<T,bits> result;
    result ^= result;
    for(int i = 0; i < 2 * bits; i ++)
      if(int(src >> i) & 1)
        result += *this << i;
    // N.B.
    //   If we work with multiply with table and summing up with simple window,
    //   and the parallel condition, we can reduce better:
    //     with bit pair [x1, x2, ..., xn], [y1, y2, ..., yn],
    //     make table [[x1*y1, ..., x1*yn],...,[xn*y1, ... xn*yn]],
    //     then counter orthogonal sum-up with parallel.
    //     we get [z1, ... zn] == [x1*y1, ..., sum_i+j=k(x_i*y_j), ..., xn*yn],
    //     then sum-up with certain bit adder and fixing one by one:
    //     r1 := x1*y1, s1 := ((x1*y1) >> 1) + z2, r2 := s2 & 1, ... and so on.
    return result;
  }
  inline DUInt<T,bits>& operator *= (const DUInt<T,bits>& src) {
    return *this = *this * src;
  }
  inline DUInt<T,bits>  operator /  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work /= src;
  }
  inline DUInt<T,bits>& operator /= (const DUInt<T,bits>& src) {
    const static DUInt<T,bits> one(int(1));
    if(! src)
      throw "Zero division";
    if(! *this)
      return *this;
    auto cache(*this);
    *this ^= *this;
    for(int i = 2 * bits - 1; 0 <= i; i --)
      if((cache >> i) >= src) {
        *this |= one << i;
        cache -= src << i;
      }
    assert(cache < src);
    return *this;
    // N.B. if we works with newton's method, better speed will be gained.
  }
  inline DUInt<T,bits>  operator %  (const DUInt<T,bits>& src) const {
    return *this - ((*this / src) * src);
  }
  inline DUInt<T,bits>& operator %= (const DUInt<T,bits>& src) {
    return *this = *this % src;
  }
  inline DUInt<T,bits>  operator << ( const int& b)            const {
    auto work(*this);
    return work <<= b;
  }
  inline DUInt<T,bits>& operator <<= (const int& b) {
    if(! b)                return *this;
    else if(b < 0)         return *this >>= (- b);
    else if(b >= bits * 2) return *this ^= *this;
    else if(b > bits) {
      e[1]  = e[0] << (b - bits);
      e[0] ^= e[0];
    } else if(b == bits) {
      e[1]  = e[0];
      e[0] ^= e[0];
    } else {
      e[1] <<= b;
      e[1]  |= e[0] >> (bits - b);
      e[0] <<= b;
    }
    return *this;
  }
  inline DUInt<T,bits>  operator >> ( const int& b)            const {
    auto work(*this);
    return work >>= b;
  }
  inline DUInt<T,bits>& operator >>= (const int& b) {
    if(! b)                return *this;
    else if(b < 0)         return *this <<= (- b);
    else if(b >= bits * 2) return *this ^= *this;
    else if(b > bits) {
      e[0]  = e[1] >> (b - bits);
      e[1] ^= e[1];
    } else if(b == bits) {
      e[0]  = e[1];
      e[1] ^= e[1];
    } else {
      e[0] >>= b;
      e[0]  |= e[1] << (bits - b);
      e[1] >>= b;
    }
    return *this;
  }
  inline DUInt<T,bits>  operator &  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work &= src;
  }
  inline DUInt<T,bits>& operator &= (const DUInt<T,bits>& src) {
    e[0] &= src.e[0]; e[1] &= src.e[1];
    return *this;
  }
  inline DUInt<T,bits>  operator |  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work |= src;
  }
  inline DUInt<T,bits>& operator |= (const DUInt<T,bits>& src) {
    e[0] |= src.e[0]; e[1] |= src.e[1];
    return *this;
  }
  inline DUInt<T,bits>  operator ^  (const DUInt<T,bits>& src) const {
    auto work(*this);
    return work ^= src;
  }
  inline DUInt<T,bits>& operator ^= (const DUInt<T,bits>& src) {
    e[0] ^= src.e[0]; e[1] ^= src.e[1];
    return *this;
  }
  inline DUInt<T,bits>  operator ~  ()                         const {
    DUInt<T,bits> work;
    work.e[0] = ~ e[0]; work.e[1] = ~ e[1];
    return work;
  }
  inline DUInt<T,bits>& operator =  (const DUInt<T,bits>& src) {
    e[0] = src.e[0]; e[1] = src.e[1];
    return *this;
  }
  inline DUInt<T,bits>& operator =  (const DUInt<DUInt<T,bits>,bits*2>& src) {
    return *this = src.e[0];
  }
  inline DUInt<T,bits>& operator =  (DUInt<T,bits>&& src) {
    e[0] = move(src.e[0]); e[1] = move(src.e[1]);
    return *this;
  }
  inline bool           operator <  (const DUInt<T,bits>& src) const {
    if(e[1]) return e[1] != src.e[1] ? e[1] < src.e[1] : e[0] < src.e[0];
    return bool(src.e[1]) || e[0] < src.e[0];
  }
  inline bool           operator <= (const DUInt<T,bits>& src) const {
    return *this < src || *this == src;
  }
  inline bool           operator >  (const DUInt<T,bits>& src) const {
    return ! (*this <= src);
  }
  inline bool           operator >= (const DUInt<T,bits>& src) const {
    return ! (*this < src);
  }
  inline bool           operator == (const DUInt<T,bits>& src) const {
    return ! (*this != src);
  }
  inline bool           operator != (const DUInt<T,bits>& src) const {
    return (*this ^ src).operator bool();
  }
  inline bool           operator && (const DUInt<T,bits>& src) const {
    return this->operator bool() && src.operator bool();
  }
  inline bool           operator || (const DUInt<T,bits>& src) const {
    return this->operator bool() || src.operator bool();
  }
  inline bool           operator !    () const {
    return ! this->operator bool();
  }
  inline                operator bool () const {
    return e[0] || e[1];
  }
  inline                operator int  () const {
    return int(e[0]);
  }
  inline                operator T    () const {
    return e[0];
  }
  inline                operator DUInt<T,bits> () const {
    return *this;
  }

  T e[2];
};

template <typename T, int bits> ostream&  operator << (ostream& os, DUInt<T,bits> v) {
  static const char* table = "0123456789abcdef";
  vector<char> buf;
  while(v) {
    buf.emplace_back(table[int(v) & 0x0f]);
    v >>= 4;
  }
  if(buf.size()) {
    for(int i = 0; 0 <= i && i < buf.size(); i ++)
      os << char(buf[buf.size() - 1 - i]);
    return os;
  }
  return os << '0';
}

template <typename T, int bits> istream&  operator >> (istream& is, DUInt<T,bits>& v) {
  v ^= v;
  // skip white spaces.
  while(! is.eof()) {
    const auto buf(is.get());
    if(buf != ' ' && buf != '\t') {
      is.unget();
      break;
    }
  }
  while(! is.eof() ) {
    const auto buf(is.get());
    if('0' <= buf && buf <= '9') {
      v <<= 4;
      v |= DUInt<T,bits>(int(buf - '0'));
    } else if('a' <= buf && buf <= 'f') {
      v <<= 4;
      v |= DUInt<T,bits>(int(buf - 'a' + 10));
    } else {
      is.unget();
      break;
    }
  }
  return is;
}


// add sign.
template <typename T, int bits> class Signed : public T {
public:
  inline Signed() { ; }
  inline Signed(const int& src) {
    T tsrc(src);
    *this = reinterpret_cast<const Signed<T,bits>&>(tsrc);
  }
  inline Signed(const T& src) {
    *this = reinterpret_cast<const Signed<T,bits>&>(src);
  }
  inline Signed(const Signed<T,bits>& src) {
    *this = src;
  }
  inline bool operator <  (const Signed<T,bits>& src) const {
    const auto mthis(int(*this >> (bits - 1)));
    const auto msrc( int(src   >> (bits - 1)));
    if(mthis ^ msrc) return mthis;
    if(mthis)
      return - dynamic_cast<const T&>(src) < - dynamic_cast<const T&>(*this);
    return dynamic_cast<const T&>(*this) < dynamic_cast<const T&>(src);
  }
  inline bool operator <= (const Signed<T,bits>& src) const {
    return ! (*this > src);
  }
  inline bool operator >  (const Signed<T,bits>& src) const {
    return ! (*this < src) && *this != src;
  }
  inline bool operator >= (const Signed<T,bits>& src) const {
    return ! (*this < src);
  }
};

template <typename T, int bits> ostream& operator << (ostream& os, Signed<T,bits> v) {
  const static Signed<T,bits> zero(0);
  if(v < zero) {
    os << '-';
    v = - v;
  }
  return os << dynamic_cast<const T&>(v);
}

// integer to integer float part.
template <typename T, typename W, int bits, typename U> class SimpleFloat {
public:
  inline SimpleFloat() {
    assert(0 < bits && ! (bits & 1));
    s |= (1 << NaN) | (1 << INF);
  }
  template <typename V> inline SimpleFloat(const V& src) {
    const static V vzero(0);
    s ^= s;
    m  = T(int(src < vzero ? - src : src));
    e ^= e;
    s |= safeAdd(e, normalize(m));
    if(src < vzero)
      s |= 1 << SIGN;
    ensureFlag();
  }
  inline SimpleFloat(const SimpleFloat<T,W,bits,U>& src) { *this = src; }
  inline SimpleFloat(SimpleFloat<T,W,bits,U>&& src) { *this = src; }
  inline ~SimpleFloat() { ; }
  
  inline SimpleFloat<T,W,bits,U>  operator -  () const {
    auto work(*this);
    work.s ^= 1 << SIGN;
    return work;
  }
  inline SimpleFloat<T,W,bits,U>  operator +  (const SimpleFloat<T,W,bits,U>& src) const {
    auto work(*this);
    return work += src;
  }
         SimpleFloat<T,W,bits,U>& operator += (const SimpleFloat<T,W,bits,U>& src) {
    if((s |= src.s & (1 << NaN)) & (1 << NaN)) return *this;
    if(s & (1 << INF)) {
      if((src.s & (1 << INF)) && (s ^ src.s) & (1 << SIGN)) s |= 1 << NaN;
      return *this;
    }
    if(src.s & (1 << INF)) return *this = src;
    if(! m) return *this = src;
    if(! src.m) return *this;
    if(! ((s ^ src.s) & (1 << SIGN))) {
      if(e >= src.e) {
        m >>= 1;
        s |= safeAdd(e, 1);
        U se(e);
        if(! safeAdd(se, - src.e) && se < U(bits))
          m += se ? src.m >> int(se) : src.m;
      } else
        return *this = src + *this;
    } else {
      if(e > src.e) {
        U se(e);
        if(! safeAdd(se, - src.e) && se < U(bits))
          m -= se ? src.m >> int(se) : src.m;
      } else if(e == src.e) {
        if(m >= src.m)
          m -= src.m;
        else
          return *this = src + *this;
      } else
        return *this = src + *this;
    }
    s |= safeAdd(e, normalize(m));
    return ensureFlag();
  }
  inline SimpleFloat<T,W,bits,U>  operator -  (const SimpleFloat<T,W,bits,U>& src) const {
    auto work(*this);
    return work -= src;
  }
  inline SimpleFloat<T,W,bits,U>& operator -= (const SimpleFloat<T,W,bits,U>& src) {
    s ^= 1 << SIGN;
    *this += src;
    s ^= 1 << SIGN;
    return *this;
  }
  inline SimpleFloat<T,W,bits,U>  operator *  (const SimpleFloat<T,W,bits,U>& src) const {
    auto work(*this);
    return work *= src;
  }
         SimpleFloat<T,W,bits,U>& operator *= (const SimpleFloat<T,W,bits,U>& src) {
    s ^= src.s & (1 << SIGN);
    if((s |= src.s & (1 << NaN)) & (1 << NaN)) return *this;
    if((! m) || (! src.m)) {
      s |= 1 << DWRK;
      return ensureFlag();
    }
    if((s |= src.s & (1 << INF)) & (1 << INF)) return *this;
    auto mm(W(m) * W(src.m));
    s |= safeAdd(e, src.e);
    s |= safeAdd(e, normalize(mm));
    s |= safeAdd(e, U(bits));
    m  = T(mm >> bits);
    return ensureFlag();
  }
  inline SimpleFloat<T,W,bits,U>  operator /  (const SimpleFloat<T,W,bits,U>& src) const {
    auto work(*this);
    return work /= src;
  }
         SimpleFloat<T,W,bits,U>& operator /= (const SimpleFloat<T,W,bits,U>& src) {
    s ^= src.s & (1 << SIGN);
    if((s |= src.s & (1 << NaN)) & (1 << NaN)) return *this;
    if(! (s & (1 << INF)) && (src.s & (1 << INF))) {
      s |= 1 << DWRK;
      return ensureFlag();
    }
    if(s & (1 << INF)) {
      if(src.s & (1 << INF)) s |= 1 << NaN;
      return *this;
    }
    if(! src.m) {
      throw "Zero division";
      s |= 1 << NaN;
      return *this;
    }
    if(! m) return *this;
    auto mm((W(m) << bits) / W(src.m));
    s |= safeAdd(e, - src.e);
    s |= safeAdd(e, normalize(mm));
    m  = T(mm >> bits);
    return ensureFlag();
  }

  inline SimpleFloat<T,W,bits,U>  operator %  (const SimpleFloat<T,W,bits,U>& src) const {
    return *this - (*this / src).floor() * src;
  }
  inline SimpleFloat<T,W,bits,U>& operator %= (const SimpleFloat<T,W,bits,U>& src) {
    return *this = *this % src;
  }
  inline SimpleFloat<T,W,bits,U>  operator <<  (const U& b) const {
    auto work(*this);
    return work <<= b;
  }
  inline SimpleFloat<T,W,bits,U>& operator <<= (const U& b) {
    if(s & ((1 << INF) | (1 << NaN))) return *this;
    s |= safeAdd(e, b);
    return ensureFlag();
  }
  inline SimpleFloat<T,W,bits,U>  operator >>  (const U& b) const {
    auto work(*this);
    return work >>= b;
  }
  inline SimpleFloat<T,W,bits,U>& operator >>= (const U& b) {
    if(s & ((1 << INF) | (1 << NaN))) return *this;
    s |= safeAdd(e, - b);
    return ensureFlag();
  }
  inline SimpleFloat<T,W,bits,U>& operator =  (const SimpleFloat<T,W,bits,U>& src) {
    s = src.s;
    e = src.e;
    m = src.m;
    return *this;
  }
  inline SimpleFloat<T,W,bits,U>& operator =  (SimpleFloat<T,W,bits,U>&& src) {
    s = move(src.s);
    e = move(src.e);
    m = move(src.m);
    return *this;
  }
  inline bool             operator == (const SimpleFloat<T,W,bits,U>& src) const {
    return ! (*this != src);
  }
  inline bool             operator != (const SimpleFloat<T,W,bits,U>& src) const {
    return (((s | src.s) & ((1 << INF) | (1 << NaN))) ||
             (s != src.s || e != src.e || m != src.m)) &&
           ! (! m && ! src.m);
  }
  inline bool             operator <  (const SimpleFloat<T,W,bits,U>& src) const {
    if((s | src.s) & (1 << NaN)) throw "compair NaN";
    const auto s_is_minus(s & (1 << SIGN));
    if(s_is_minus ^ (src.s & (1 << SIGN))) return s_is_minus;
    if(s & (1 << INF)) {
      if(src.s & (1 << INF)) throw "compair INF";
      return s_is_minus;
    }
    if(src.s & (1 << INF)) return ! s_is_minus;
    if(m && src.m) {
      if(e < src.e) return ! s_is_minus;
      if(e == src.e) return s_is_minus ? src.m < m : m < src.m;
      return s_is_minus;
    }
    return !m ? (! src.m ? m != src.m : ! (src.s & (1 << SIGN))) : s_is_minus;
  }
  inline bool             operator <= (const SimpleFloat<T,W,bits,U>& src) const {
    return *this < src || *this == src;
  }
  inline bool             operator >  (const SimpleFloat<T,W,bits,U>& src) const {
    return ! (*this <= src);
  }
  inline bool             operator >= (const SimpleFloat<T,W,bits,U>& src) const {
    return ! (*this < src);
  }
  inline bool             operator !  () const {
    return ! m && isfinite(*this);
  }
  inline                  operator bool () const {
    return ! (!*this);
  }
  inline                  operator int  () const {
    return int(this->operator T());
  }
  inline                  operator T    () const {
    auto deci(*this);
    if(deci.s & (1 << INF)) throw "Inf to convert int";
    if(deci.s & (1 << NaN)) throw "NaN to convert int";
    if(! deci.m) return T(int(0));
    if(U(bits) <= deci.e || (uzero() < deci.e && (deci.m << int(deci.e)) >> int(deci.e) != deci.m))
      throw "Overflow to convert int.";
    if(deci.e <= - U(bits)) return T(int(0));
    if(     deci.e < uzero()) deci.m >>= - int(deci.e);
    else if(uzero() < deci.e) deci.m <<=   int(deci.e);
    return s & (1 << SIGN) ? - deci.m : deci.m;
  }
  inline                  operator SimpleFloat<T,W,bits,U> () const {
    return *this;
  }
  // XXX: absfloor, absceil implementation.
  inline SimpleFloat<T,W,bits,U>  floor() const {
    if(uzero() <= e) return *this;
    if(e <= - U(bits)) return zero();
    auto deci(*this);
    deci.m >>= - int(deci.e);
    deci.m <<= - int(deci.e);
    return deci;
  }
  inline SimpleFloat<T,W,bits,U>  ceil() const {
    const auto fl(this->floor());
    if(*this - fl) {
      auto pmone(one());
      pmone.s |= s & (1 << SIGN);
      return fl + pmone;
    }
    return fl;
  }

  inline SimpleFloat<T,W,bits,U>  abs()  const {
    auto work(*this);
    work.s &= ~ (1 << SIGN);
    return work;
  }
         SimpleFloat<T,W,bits,U>  log()  const;
         SimpleFloat<T,W,bits,U>  exp()  const;
         SimpleFloat<T,W,bits,U>  sin()  const;
         SimpleFloat<T,W,bits,U>  cos()  const;
         SimpleFloat<T,W,bits,U>  atan() const;
  inline SimpleFloat<T,W,bits,U>  sqrt() const;
  
  unsigned char s;
  typedef enum {
    INF = 0,
    NaN = 1,
    SIGN = 2,
    DWRK = 3
  } state_t;
  T m;
  U e;
  const U& uzero() const {
    const static U vuzero(0);
    return vuzero;
  }
  const SimpleFloat<T,W,bits,U>& zero()   const {
    const static SimpleFloat<T,W,bits,U> vzero(0);
    return vzero;
  }
  const SimpleFloat<T,W,bits,U>& one()    const {
    const static SimpleFloat<T,W,bits,U> vone(1);
    return vone;
  }
  const SimpleFloat<T,W,bits,U>& two()    const {
    const static auto vtwo(one() << U(1));
    return vtwo;
  }
  const SimpleFloat<T,W,bits,U>& pi()     const {
    const static auto vpi(quatpi() << U(2));
    return vpi;
  }
  const SimpleFloat<T,W,bits,U>& halfpi() const {
    const static auto vhalfpi(quatpi() << U(1));
    return vhalfpi;
  }
  const SimpleFloat<T,W,bits,U>& quatpi() const {
    const static auto vquatpi(one().atan());
    return vquatpi;
  }
  const SimpleFloat<T,W,bits,U>& twopi()  const {
    const static auto vtwopi(quatpi() << U(3));
    return vtwopi;
  }
  const SimpleFloat<T,W,bits,U>& sqrt2()  const {
    const static auto vsqrt2((one() << U(1)).sqrt());
    return vsqrt2;
  }
private:
  template <typename V> inline U normalize(V& src) const {
    V   bt(int(1));
    int b(0);
    int tb(0);
    for( ; bt; tb ++) {
      if(src & bt) b = tb;
      bt <<= 1;
    }
    const auto shift(tb - b - 1);
    assert(0 <= shift);
    if(shift) src <<= shift;
    return - U(shift);
  }
  inline SimpleFloat<T,W,bits,U>& ensureFlag() {
    if(s & (1 << INF))
      s &= ~ (1 << DWRK);
    else if(! m || (s & (1 << DWRK))) {
      e ^= e;
      m ^= m;
      s &= ~ ((1 << DWRK) | (1 << INF));
    }
    return * this;
  }
  inline unsigned char safeAdd(U& dst, const U& src) {
    const auto dst0(dst);
    dst += src;
    if((dst0 > uzero() && src > uzero() && dst <= uzero()) ||
       (dst0 < uzero() && src < uzero() && dst >= uzero()))
      return 1 << (dst0 < uzero() ? DWRK : INF);
    return 0;
  }
  inline char residue2() const {
    if(uzero() < e || U(bits) <= - e) return 0;
    if(! e) return char(int(m) & 1);
    return char(int(m >> - int(e)) & 1);
  }

  // XXX: these are NOT threadsafe on first call.
  const vector<SimpleFloat<T,W,bits,U> >& exparray()    const;
  const vector<SimpleFloat<T,W,bits,U> >& invexparray() const;
/*
friend:
  ostream&    operator << (ostream& os, const SimpleFloat<T,W,bits,U>& v);
  istream&    operator >> (istream& is, SimpleFloat<T,W,bits,U>& v);
*/
};

template <typename T, typename W, int bits, typename U> SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::log() const {
  const static auto einv(one() / one().exp());
  const static auto one_einv(one() + einv);
  if((s & (1 << SIGN)) && m)
    throw "Negative log";
  if(s & ((1 << INF) | (1 << NaN)))
    return *this;
  if(! m) {
    auto work(*this);
    work.s |= (1 << INF) | (1 << SIGN);
    return work;
  }
  if(einv <= *this && *this <= one_einv) {
    // ln(x) = (x - 1) - (x - 1)^2/2 + (x-1)^3/3- ...
    const auto dx(*this - one());
          auto x(dx);
          auto before(one());
          auto res(zero());
    for(int t = 1; (res - before).m; t ++, x *= dx) {
      const auto abst(x / SimpleFloat<T,W,bits,U>(t));
      before = res;
      res   += (t % 2 ? abst : - abst);
    }
    return res;
  }
  static const auto& ea(exparray());
  static const auto& iea(invexparray());
        auto  result(zero());
        auto  work(*this);
  if(one_einv < work) {
    for(int i = min(ea.size(), iea.size()) - 1; 0 < i; i --)
      if(ea[i] <= work) {
        result += one() << U(i - 1);
        work   *= iea[i];
      }
    if(! (work <= one_einv)) {
      result += one();
      work   *= iea[1];
    }
  } else {
    for(int i = min(ea.size(), iea.size()) - 1; 0 < i; i --)
      if(work <= iea[i]) {
        result -= one() << U(i - 1);
        work   *= ea[i];
      }
    if(! (einv <= work)) {
      result -= one();
      work   *= ea[1];
    }
  }
  assert(einv <= work && work <= one_einv);
  return result += work.log();
}

template <typename T, typename W, int bits, typename U> SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::exp() const {
  if(s & ((1 << INF) | (1 << NaN))) {
    if(! (s & (1 << NaN)) && (s & (1 << SIGN)))
      return zero();
    return *this;
  }
  if(this->abs() <= one()) {
    // exp(x) = 1 + x/1! + x^2/2! + ...
    auto denom(one());
    auto x(*this);
    auto before(zero());
    auto res(one());
    for(int t = 1; (res - before).m; t ++, x *= *this) {
      before = res;
      denom *= SimpleFloat<T,W,bits,U>(t);
      res   += x / denom;
    }
    return res;
  }
  static const auto& en(exparray());
  static const auto& ien(invexparray());
        auto  work(this->abs());
        auto  result(one());
  for(int i = 1; 0 <= i && i < min(en.size(), ien.size()) && work.floor(); i ++, work >>= U(1))
    if(work.residue2())
      result *= s & (1 << SIGN) ? ien[i] : en[i];
  if(work.floor()) {
    work.s |= 1 << INF;
    return work;
  }
  const auto residue(*this - this->floor());
  assert(residue.abs() <= one());
  return result *= residue.exp();
}

template <typename T, typename W, int bits, typename U> SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::sin() const {
  if(s & ((1 << INF) | (1 << NaN))) {
    auto res(*this);
    res.s |= 1 << NaN;
    return res;
  }
  if(- one() <= *this && *this <= one()) {
    // sin(x) = x - x^3/3! + x^5/5! - ...
    const auto sqx(*this * *this);
          auto denom(one());
          auto x(sqx * *this);
          auto before(zero());
          auto res(*this);
    for(int t = 1; (res - before).m; t ++, x *= sqx) {
      SimpleFloat<T,W,bits,U> tt(t);
      tt   <<= U(1);
      before = res;
      denom *= - tt * (tt + one());
      res   += x / denom;
    }
    return res;
  }
  if(- halfpi() <= *this && *this <= halfpi())
    return ((*this - quatpi()).cos() + (*this - quatpi()).sin()) / sqrt2();
  if(this->abs() == pi())
    return zero();
  if(- pi() <= *this && *this <= pi())
    return (halfpi() - *this).cos();
  if(- twopi() <= *this && *this <= twopi())
    return - (*this + pi()).sin();
  return (*this % twopi()).sin();
}

template <typename T, typename W, int bits, typename U> SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::cos() const {
  if(s & ((1 << INF) | (1 << NaN))) {
    auto res(*this);
    res.s |= 1 << NaN;
    return res;
  }
  if(- one() <= *this && *this <= one()) {
    // cos(x) = 1 - x^2/2! + x^4/4! - ...
    const auto sqx(*this * *this);
          auto denom(one());
          auto x(sqx);
          auto before(zero());
          auto res(one());
    for(int t = 1; (res - before).m; t ++, x *= sqx) {
      SimpleFloat<T,W,bits,U> tt(t);
      tt   <<= U(1);
      before = res;
      denom *= - tt * (tt - one());
      res   += x / denom;
    }
    return res;
  }
  if(this->abs() == halfpi())
    return zero();
  if(- halfpi() <= *this && *this <= halfpi())
    return ((*this - quatpi()).cos() - (*this - quatpi()).sin()) / sqrt2();
  if(this->abs() == pi())
    return - one();
  if(- pi() <= *this && *this <= pi())
    return (halfpi() - *this).sin();
  if(- twopi() <= *this && *this <= twopi())
    return - (*this + pi()).cos();
  return (*this % twopi()).cos();
}

template <typename T, typename W, int bits, typename U> SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::atan() const {
  if(s & ((1 << INF) | (1 << NaN))) {
    if(! (s & (1 << NaN)))
      return s & (1 << SIGN) ? - halfpi() : halfpi();
    return *this;
  }
  if(s & (1 << SIGN))
    return - (- *this).atan();
  static const auto half(one() >> U(1));
  static const auto four(one() << U(2));
  static const auto five((one() << U(2)) + one());
  if(- half <= *this && *this <= half) {
    // arctan(x) = x - x^3/3 + x^5/5 - ...
    const auto sqx(*this * *this);
          auto x(sqx * *this);
          auto before(zero());
          auto res(*this);
    for(int t = 1; (res - before).m; t ++, x *= sqx) {
      const auto abst(x / ((SimpleFloat<T,W,bits,U>(t) << U(1)) + one()));
      before = res;
      res   += (t % 2 ? - abst : abst);
    }
    return res;
  }
  // N.B.
  //  atan(u) + atan(v) = atan((u + v) / (1 - uv)) mod pi, uv != 1.
  //    in u = 0.5, v = x - 0.5 case,
  //  atan(x / (1 - x / 2 + 1 / 4)) = atan(.5) + atan(x - .5) =
  //  atan(x / (1.25 - .5 * x)) 
  //  y := x / (1.25 - .5 * x) then,
  //  (1.25 - .5 * x) * y = x,
  //  (5 - 2x) * y = 4 x
  //  x = 5y / (4 + 2y),
  //     y - x = ((4 + 2y) * y - 5y) / (4 + 2y)
  //           = y * (2y - 1) / (4 + 2y)
  //     so 0 <= y and 0 < y case, this makes decreasing function.
  //       (v = x - .5 and 0 <= 2y - 1)
  if(- two() <= *this && *this <= two()) {
    static const auto atanhalf(half.atan());
    const auto v(five * *this / (four + (*this << U(1))) - half);
    assert(v < *this);
    return atanhalf + v.atan();
  }
  // N.B.
  //    in u = v case,
  //  2 atan(u) = atan(2 * u / (1 - u * u))
  //  2 atan(u) = atan(2 * u / (1 - u) / (1 + u))
  //            = atan(2 / (1 / u - u))
  //    in 2Y := 1 / u - u case,
  //            = atan(1 / Y),
  //  u^2 + 2Yu - 1 == 0, u = - Y \pm sqrt(Y^2 + 1)
  const auto Y(one() / (*this));
  const auto u((Y * Y + one()).sqrt() - Y);
  assert(- *this < u && u < *this);
  return u.atan() << U(1);
}

template <typename T, typename W, int bits, typename U> const vector<SimpleFloat<T,W,bits,U> >& SimpleFloat<T,W,bits,U>::exparray() const {
  static vector<SimpleFloat<T,W,bits,U> > ebuf;
  if(ebuf.size())
    return ebuf;
  ebuf.emplace_back(one());
  ebuf.emplace_back(ebuf[0].exp());
  for(int i = 1; 0 <= i; i ++) {
    const auto en(ebuf[i] * ebuf[i]);
    if(en && isfinite(en))
      ebuf.emplace_back(en);
    else
      break;
  }
  return ebuf;
}

template <typename T, typename W, int bits, typename U> const vector<SimpleFloat<T,W,bits,U> >& SimpleFloat<T,W,bits,U>::invexparray() const {
  static vector<SimpleFloat<T,W,bits,U> > iebuf;
  if(iebuf.size())
    return iebuf;
  const auto& ea(exparray());
  for(int i = 0; 0 <= i && i < ea.size(); i ++) {
    const auto ien(one() / ea[i]);
    if(ien && isfinite(ien))
      iebuf.emplace_back(ien);
    else
      break;
  }
  return iebuf;
}

template <typename T, typename W, int bits, typename U> inline SimpleFloat<T,W,bits,U> SimpleFloat<T,W,bits,U>::sqrt() const {
  if(s & ((1 << INF) | (1 << NaN))) {
    auto res(*this);
    if(s & (1 << SIGN)) res.s |= 1 << NaN;
    return res;
  }
  auto res((this->log() >> U(1)).exp());
  // get better accuracy (is this enough?, double accuracy on one loop.)
  // newton's method: 0 == f'(x_n) (x_{n+1} - x_n) + f(x_n)
  //            x_{n+1} := x_n - f(x_n)/f'(x_n).
  //         where f(x) := x_n * x_n - *this
  if(! res)
    return res;
  return (res + *this / res) >> U(1);
}

template <typename T, typename W, int bits, typename U> ostream& operator << (ostream& os, const SimpleFloat<T,W,bits,U>& v) {
  static const U uzero(int(0));
  if(isnan(v))
    return os << "NaN ";
  if(isinf(v))
    return os << (const char*)(v.s & (1 << v.SIGN) ? "-" : "") << "Inf ";
  return os << (const char*)(v.s & (1 << v.SIGN) ? "-" : "") << std::hex << T(v.m) << "*2^" << (const char*)(v.e < uzero ? "-" : "") << (v.e < uzero ? U(- v.e) : v.e) << " " << std::dec;
}

template <typename T, typename W, int bits, typename U> istream& operator >> (istream& is, SimpleFloat<T,W,bits,U>& v) {
  const static SimpleFloat<T,W,bits,U> two(2);
               SimpleFloat<T,W,bits,U> e(0);
  bool mode(false);
  bool sign(false);
  bool fsign(false);
  v = SimpleFloat<T,W,bits,U>(0);
  // skip white spaces.
  while(! is.eof()) {
    const auto buf(is.get());
    if(buf != ' ' && buf != '\t' && buf != '\n') {
      is.unget();
      break;
    }
  }
  while(! is.eof() && ! is.bad() ) {
    const auto buf(is.get());
    switch(buf) {
    case '-':
      sign  = true;
    case '+':
      if(fsign)
        throw "Wrong input";
      fsign = true;
      break;
    case '*':
      if(mode)
        goto ensure;
      if(sign)
        v   = - v;
      mode  = true;
      sign  = false;
      fsign = false;
      if(is.get() != '2') {
        is.unget();
        goto ensure;
      }
      if(is.get() != '^') {
        is.unget();
        is.unget();
        goto ensure;
      }
      break;
    case '.':
      throw "not implemented now";
      break;
    case '0': case '1': case '2': case '3': case '4':
    case '5': case '6': case '7': case '8': case '9':
      if(mode) {
        e <<= U(int(4));
        e  += SimpleFloat<T,W,bits,U>(int(buf - '0'));
      } else {
        v <<= U(int(4));
        v  += SimpleFloat<T,W,bits,U>(int(buf - '0'));
      }
      fsign = true;
      break;
    case 'a': case'b': case 'c': case 'd': case 'e': case 'f':
      if(mode) {
        e <<= U(int(4));
        e  += SimpleFloat<T,W,bits,U>(int(buf - 'a' + 10));
      } else {
        v <<= U(int(4));
        v  += SimpleFloat<T,W,bits,U>(int(buf - 'a' + 10));
      }
      fsign = true;
      break;
    default:
      goto ensure;
    }
  }
 ensure:
  if(sign) {
    if(mode)
      e = - e;
    else
      v = - v;
  }
  v *= pow(two, e);
  return is;
}

template <typename T, typename W, int bits, typename U> static inline bool isinf(const SimpleFloat<T,W,bits,U>& src) {
  return src.s & (1 << src.INF);
}

template <typename T, typename W, int bits, typename U> static inline bool isnan(const SimpleFloat<T,W,bits,U>& src) {
  return src.s & (1 << src.NaN);
}

template <typename T, typename W, int bits, typename U> static inline bool isfinite(const SimpleFloat<T,W,bits,U>& src) {
  return ! (src.s & ((1 << src.INF) | (1 << src.NaN)));
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> floor(const SimpleFloat<T,W,bits,U>& src) {
  return src.floor();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> ceil(const SimpleFloat<T,W,bits,U>& src) {
  return src.ceil();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> abs(const SimpleFloat<T,W,bits,U>& src) {
  return src.abs();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> sqrt(const SimpleFloat<T,W,bits,U>& src) {
  return src.sqrt();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> exp(const SimpleFloat<T,W,bits,U>& src) {
  return src.exp();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> log(const SimpleFloat<T,W,bits,U>& src) {
  return src.log();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> sin(const SimpleFloat<T,W,bits,U>& src) {
  return src.sin();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> cos(const SimpleFloat<T,W,bits,U>& src) {
  return src.cos();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> tan(const SimpleFloat<T,W,bits,U>& src) {
  return src.sin() / src.cos();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> atan(const SimpleFloat<T,W,bits,U>& src) {
  return src.atan();
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> atan2(const SimpleFloat<T,W,bits,U>& y, const SimpleFloat<T,W,bits,U>& x) {
  auto atan0(y.halfpi());
  if(! x && ! y)
    return x / y;
  else if(isfinite(x)) {
    if(! isfinite(y) )
      goto ensure;
    if(! x)
      goto ensure;
    const auto yoverx((y / x).abs());
    if(! isfinite(yoverx) )
      goto ensure;
    const auto atan00(yoverx.atan());
    if(! isfinite(atan00) )
      goto ensure;
    atan0 = atan00;
    goto ensure;
  } else if(isfinite(y)) {
    atan0 = x.zero();
    goto ensure;
  }
  return y;
 ensure:
  if(y.s & (1 << y.SIGN)) {
    if(x.s & (1 << x.SIGN))
      atan0 = - (x.pi() - atan0);
    else
      atan0 = - atan0;
  } else if(x.s & (1 << x.SIGN))
    atan0  = x.pi() - atan0;
  return atan0;
}

template <typename T, typename W, int bits, typename U> static inline SimpleFloat<T,W,bits,U> pow(const SimpleFloat<T,W,bits,U>& src, const SimpleFloat<T,W,bits,U>& dst) {
  if(! dst) {
    if(! src)
      throw "0^0";
    return dst.one();
  }
  return exp(log(src) * dst);
}


// class complex part:
template <typename T> class Complex {
public:
  inline Complex() { ; }
  inline Complex(const T& real, const T& imag = T(int(0))) {
    _real = real; _imag = imag;
  }
  inline Complex(const Complex<T>& s) { *this = s; }
  inline Complex(Complex<T>&& s) { *this = s; }
  inline Complex(T&& real) {
    const static T zero(0);
    _real = move(real);
    _imag = zero;
  }
  inline Complex(T&& real, T&& imag) {
    _real = move(real);
    _imag = move(imag);
  }
  inline ~Complex() { ; }

  inline Complex<T>  operator ~  ()                    const {
    return Complex<T>(  _real, - _imag);
  }
  inline Complex<T>  operator -  ()                    const {
    return Complex<T>(- _real, - _imag);
  }
  inline Complex<T>  operator +  (const Complex<T>& s) const {
    auto result(*this);
    return result += s;
  }
  inline Complex<T>& operator += (const Complex<T>& s) {
    _real += s._real;
    _imag += s._imag;
    return *this;
  }
  inline Complex<T>  operator -  (const Complex<T>& s) const {
    auto result(*this);
    return result -= s;
  }
  inline Complex<T>& operator -= (const Complex<T>& s) {
    _real -= s._real;
    _imag -= s._imag;
    return *this;
  }
  inline Complex<T>  operator *  (const T& s)          const {
    auto result(*this);
    return result *= s;
  }
  inline Complex<T>& operator *= (const T& s) {
    _real *= s;
    _imag *= s;
    return *this;
  }
  inline Complex<T>  operator *  (const Complex<T>& s) const {
    return Complex<T>(_real * s._real - _imag * s._imag,
                      _real * s._imag + _imag * s._real);
  }
  inline Complex<T>& operator *= (const Complex<T>& s) {
    return (*this) = (*this) * s;
  }
  inline Complex<T>  operator /  (const T& s)          const {
    auto result(*this);
    return result /= s;
  }
  inline Complex<T>& operator /= (const T& s) {
    _real /= s;
    _imag /= s;
    return *this;
  }
  inline Complex<T>  operator /  (const Complex<T>& s) const {
    return (*this * (~ s)) / (s._real * s._real + s._imag * s._imag);
  }
  inline Complex<T>& operator /= (const Complex<T>& s) {
    return *this = *this / s;
  }
  inline bool        operator == (const Complex<T>& s) const {
    return !(*this != s);
  }
  inline bool        operator != (const Complex<T>& s) const {
    return (_real != s._real) || (_imag != s._imag);
  }
  inline bool        operator !  ()                    const {
    return !_real && !_imag;
  }
  inline Complex<T>  operator &  (const Complex<T>& s) const {
    auto result(*this);
    return result &= s;
  }
  inline Complex<T>& operator &= (const Complex<T>& s) {
    _real &= s._real;
    _imag &= s._imag;
    return *this;
  }
  inline Complex<T>  operator |  (const Complex<T>& s) const {
    auto result(*this);
    return result |= s;
  }
  inline Complex<T>& operator |= (const Complex<T>& s) {
    _real |= s._real;
    _imag |= s._imag;
    return *this;
  }
  inline Complex<T>  operator ^  (const Complex<T>& s) const {
    auto result(*this);
    return result ^= s;
  }
  inline Complex<T>& operator ^= (const Complex<T>& s) {
    _real ^= s._real;
    _imag ^= s._imag;
    return *this;
  }
  inline bool        operator && (const Complex<T>& s) const {
    return *this && s;
  }
  inline bool        operator || (const Complex<T>& s) const {
    return *this || s;
  }
  inline Complex<T>& operator =  (const Complex<T>& s) {
    _real = s._real;
    _imag = s._imag;
    return *this;
  }
  inline Complex<T>& operator =  (Complex<T>&& s) {
    _real = move(s._real);
    _imag = move(s._imag);
    return *this;
  }
  inline T&          operator [] (const size_t& i) {
    assert(0 <= i && i < 2);
    if(i) return _imag;
    return _real;
  }
  inline             operator bool () const {
    return ! (! *this);
  }
  inline             operator T    () const {
    return this->_real;
  }
  
  const Complex<T>& i() const {
    const static auto I(Complex<T>(T(int(0)), T(int(1))));
    return I;
  }
  inline T  abs() const {
    return sqrt(_real * _real + _imag * _imag);
  }
  inline T  arg() const {
    return atan2(_imag, _real);
  }
  inline T& real() {
    return _real;
  }
  inline T& imag() {
    return _imag;
  }
  inline const T& real() const {
    return _real;
  }
  inline const T& imag() const {
    return _imag;
  }
  T _real;
  T _imag;
};

template <typename T> ostream& operator << (ostream& os, const Complex<T>& v) {
  return os << v.real() << "+i" << v.imag();
}

template <typename T> istream& operator >> (istream& is, Complex<T>& v) {
  is >> v._real;
  if('+' != is.get()) {
    is.unget();
    goto ensure;
  }
  if('i' != is.get()) {
    is.unget();
    is.unget();
    goto ensure;
  }
  is >> v._imag;
  return is;
 ensure:
  v._imag = T(int(0));
  return is;
}


template <typename T> static inline T abs(const Complex<T>& s) {
  return s.abs();
}

template <typename T> static inline T arg(const Complex<T>& s) {
  return s.arg();
}

template <typename T> static inline const T& real(const Complex<T>& s) {
  return s.real();
}

template <typename T> static inline const T& imag(const Complex<T>& s) {
  return s.imag();
}

template <typename T> static inline Complex<T> exp(const Complex<T>& s) {
  return Complex<T>(exp(s.real())) * Complex<T>(cos(s.imag()), sin(s.imag()));
}

template <typename T> static inline Complex<T> log(const Complex<T>& s) {
  // N.B. main branch
  return Complex<T>(log(abs(s)), arg(s));
}

template <typename T> static inline Complex<T> sqrt(const Complex<T>& s) {
  return exp(log(s) * Complex<T>(T(int(1)) / T(int(2))));
}

template <typename T> static inline Complex<T> csin(const Complex<T>& s) {
  return (exp(Complex<T>(T(int(0)), s)) - exp(Complex<T>(T(int(0)), - s))) / Complex<T>(T(int(0)), T(int(2)));
}

template <typename T> static inline Complex<T> ccos(const Complex<T>& s) {
  return (exp(Complex<T>(T(int(0)), s)) + exp(Complex<T>(T(int(0)), - s))) / T(int(2));
}

template <typename T> static inline Complex<T> ctan(const Complex<T>& s) {
  return csin(s) / ccos(s);
}

template <typename T> static inline Complex<T> ccsc(const Complex<T>& s) {
  return Complex<T>(T(int(1))) / csin(s);
}

template <typename T> static inline Complex<T> csec(const Complex<T>& s) {
  return Complex<T>(T(int(1))) / ccos(s);
}

template <typename T> static inline T ccot(const T& s) {
  return Complex<T>(T(int(1))) / ctan(s);
}

template <typename T> using complex = Complex<T>;

#if !defined(_FLOAT_BITS_)
  #include <cmath>
  using namespace std;
  typedef uint64_t myuint;
  typedef int64_t  myint;
  // XXX:
  typedef long double myfloat;
  //typedef double myfloat;
#elif _FLOAT_BITS_ == 8
  typedef uint8_t myuint;
  typedef int8_t  myint;
  typedef SimpleFloat<myuint, uint16_t, 8, myint> myfloat;
#elif _FLOAT_BITS_ == 16
  typedef uint16_t myuint;
  typedef int16_t  myint;
  typedef SimpleFloat<myuint, uint32_t, 16, myint> myfloat;
#elif _FLOAT_BITS_ == 32
  typedef uint32_t myuint;
  typedef int32_t  myint;
  typedef SimpleFloat<myuint, uint64_t, 32, myint> myfloat;
#elif _FLOAT_BITS_ == 64
  typedef uint64_t myuint;
  typedef int64_t  myint;
  typedef SimpleFloat<myuint, unsigned __int128, 64, myint> myfloat;
#elif _FLOAT_BITS_ == 128
  typedef DUInt<uint64_t, 64> uint128_t;
  typedef Signed<uint128_t, 128> int128_t;
  typedef uint128_t myuint;
  typedef int128_t  myint;
  typedef SimpleFloat<myuint, DUInt<myuint, 128>, 128, myint> myfloat;
#elif _FLOAT_BITS_ == 256
  typedef DUInt<uint64_t, 64> uint128_t;
  typedef DUInt<uint128_t, 128> uint256_t;
  typedef Signed<uint256_t, 256> int256_t;
  typedef uint256_t myuint;
  typedef int256_t  myint;
  typedef SimpleFloat<myuint, DUInt<myuint, 256>, 256, myint> myfloat;
#elif _FLOAT_BITS_ == 512
  typedef DUInt<uint64_t, 64> uint128_t;
  typedef DUInt<uint128_t, 128> uint256_t;
  typedef DUInt<uint256_t, 256> uint512_t;
  typedef Signed<uint512_t, 512> int512_t;
  typedef uint512_t myuint;
  typedef int512_t  myint;
  typedef SimpleFloat<myuint, DUInt<myuint, 512>, 512, myint> myfloat;
#elif _FLOAT_BITS_ == 1024
  typedef DUInt<uint64_t, 64> uint128_t;
  typedef DUInt<uint128_t, 128> uint256_t;
  typedef DUInt<uint256_t, 256> uint512_t;
  typedef DUInt<uint512_t, 512> uint1024_t;
  typedef Signed<uint1024_t, 1024> int1024_t;
  typedef uint1024_t myuint;
  typedef int1024_t  myint;
  typedef SimpleFloat<myuint, DUInt<myuint, 1024>, 1024, myint> myfloat;
#else
# error cannot handle float
#endif


// N.B. start simplelin.
template <typename T> class SimpleVector {
public:
  inline SimpleVector() { ; }
  inline SimpleVector(const int& size) {
    assert(0 <= size);
    this->entity.resize(size);
  }
  inline SimpleVector(const SimpleVector<T>& other) { *this = other; }
  inline SimpleVector(SimpleVector<T>&& other) { *this = other; }
  inline ~SimpleVector() { ; }
  
  inline       SimpleVector<T>  operator -  () const {
    SimpleVector<T> res(entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      res.entity[i] = - entity[i];
    return res;
  }
  inline       SimpleVector<T>  operator +  (const SimpleVector<T>& other) const {
    auto res(*this);
    return res += other;
  }
  inline       SimpleVector<T>& operator += (const SimpleVector<T>& other) {
    assert(entity.size() == other.entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] += other.entity[i];
    return *this;
  }
  inline       SimpleVector<T>  operator -  (const SimpleVector<T>& other) const {
    auto res(*this);
    return res -= other;
  }
  inline       SimpleVector<T>& operator -= (const SimpleVector<T>& other) {
    return *this += - other;
  }
  inline       SimpleVector<T>  operator *  (const T& other) const {
    auto res(*this);
    return res *= other;
  }
  inline       SimpleVector<T>& operator *= (const T& other) {
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] *= other;
    return *this;
  }
  inline       SimpleVector<T>  operator /  (const T& other) const {
    auto res(*this);
    return res /= other;
  }
  inline       SimpleVector<T>& operator /= (const T& other) {
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] /= other;
    return *this;
  }
  inline       SimpleVector<T>& operator =  (const SimpleVector<T>& other) {
    entity = other.entity;
    return *this;
  }
  inline       SimpleVector<T>& operator =  (SimpleVector<T>&& other) {
    entity = move(other.entity);
    return *this;
  }
  inline       bool             operator == (const SimpleVector<T>& other) const {
    return ! (*this != other);
  }
  inline       bool             operator != (const SimpleVector<T>& other) const {
    if(entity.size() != other.entity.size()) return true;
    for(int i = 0; i < entity.size(); i ++)
      if(entity[i] != other.entity[i]) return true;
    return false;
  }
  template <typename U> inline T dot(const SimpleVector<U>& other) const {
    assert(entity.size() == other.entity.size());
    SimpleVector<T> work(other.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      work[i] = entity[i] * other.entity[i];
    auto res(work[0]);
    for(int i = 1; i < entity.size(); i ++)
      res += work[i];
    return res;
  }
  inline       T&               operator [] (const int& idx) {
    assert(0 <= idx && idx < entity.size());
    return entity[idx];
  }
  inline const T&               operator [] (const int& idx) const {
    assert(0 <= idx && idx < entity.size());
    return entity[idx];
  }
  template <typename U> inline SimpleVector<U> real() const {
    SimpleVector<U> result(entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      result.entity[i] = U(entity[i].real());
    return result;
  }
  template <typename U> inline SimpleVector<U> imag() const {
    SimpleVector<U> result(entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      result.entity[i] = U(entity[i].imag());
    return result;
  }
  template <typename U> inline SimpleVector<U> cast() const {
    SimpleVector<U> result(entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < entity.size(); i ++)
      result.entity[i] = U(entity[i]);
    return result;
  }
  inline const int size() const {
    return entity.size();
  }
  inline       void resize(const int& size) {
    assert(0 <= size);
    entity.resize(size);
    return;
  }
  inline       SimpleVector<T>  subVector(const int& i, const int& s) const {
    assert(0 <= s && 0 <= i && i + s <= size());
    SimpleVector<T> res(s);
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int ii = i; ii < i + s; ii ++)
      res[ii - i] = (*this)[ii];
    return res;
  }
  inline       SimpleVector<T>& setVector(const int& i, const SimpleVector<T>& d) {
    assert(0 <= i && i + d.size() <= size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int ii = i; ii < i + d.size(); ii ++)
      (*this)[ii] = d[ii - i];
    return *this;
  }
  inline       SimpleVector<T>& O(const T& r = T(int(0))) {
    return I(r);
  }
  inline       SimpleVector<T>& I(const T& r = T(int(1))) {
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < size(); i ++)
      (*this)[i] = r;
    return *this;
  }
  inline       SimpleVector<T>& ek(const int& i, const T& r = T(int(1))) {
    static const T zero(0);
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int ii = 0; ii < size(); ii ++)
      (*this)[ii] = ii == i ? r : zero;
    return *this;
  }
  inline       SimpleVector<T>  reverse() {
    SimpleVector<T> res(entity.size());
#if defined(_OPENMP)
#pragma omp simd
#endif
    for(int i = 0; i < res.size(); i ++)
      res[i] = entity[entity.size() - 1 - i];
    return res;
  }
  
  vector<T> entity;
};

template <typename T> ostream& operator << (ostream& os, const SimpleVector<T>& v) {
  SimpleVector<string> buf(v.size());
  int M(0);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < v.size(); i ++) {
    stringstream ss;
    ss << v[i];
    buf[i] = ss.str();
#if defined(_OPENMP)
#pragma omp critical
#endif
    {
      M = max(int(buf[i].size()), M);
    }
  }
  os << v.size() << " : [";
  for(int i = 0; i < buf.size(); i ++) {
    for(int j = buf[i].size(); j <= M; j ++)
      os << " ";
    os << buf[i];
    if(i < buf.size() - 1) os << ", ";
  }
  os << "]" << endl;
  return os;
}

template <typename T> istream& operator >> (istream& is, SimpleVector<T>& v) {
  int s;
  is >> s;
  if(s <= 0) return is;
  v.resize(s);
  int i(0);
  for( ; i < v.size() && ! is.eof() && ! is.bad(); ) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == ':' || c == ',' || c == '[' || c == '\n') continue;
    is.unget();
    is >> v[i ++];
  }
  while(!is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == '\n') continue;
    else if(c == ']') break;
    is.unget();
    cerr << "XXX SimpleVector<T>::operator >> (\']\')" << flush;
    break;
  }
  while(!is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t') continue;
    else if(c == '\n') break;
    is.unget();
    cerr << "XXX SimpleVector<T>::operator >> (\'\\n\')" << flush;
    break;
  }
  if(i < v.size()) {
    cerr << "XXX SimpleVector<T>::operator >> (index)" << flush;
    for( ; i < v.size(); i ++)
      v[i] = T(int(0));
  }
  return is;
}


template <typename T> class SimpleMatrix {
public:
  inline SimpleMatrix() { ecols = 0; }
  inline SimpleMatrix(const int& rows, const int& cols) {
    assert(0 <= rows && 0 <= cols);
    entity.resize(rows);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i].resize(cols);
    ecols = cols;
  }
  inline SimpleMatrix(const SimpleMatrix<T>& other) { *this = other; }
  inline SimpleMatrix(SimpleMatrix<T>&& other) { *this = other; }
  inline ~SimpleMatrix() { ; }
  
  inline       SimpleMatrix<T>  operator -  () const {
    SimpleMatrix<T> res(entity.size(), ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      res.entity[i] = - entity[i];
    return res;
  }
  inline       SimpleMatrix<T>  operator +  (const SimpleMatrix<T>& other) const {
    auto res(*this);
    return res += other;
  }
  inline       SimpleMatrix<T>& operator += (const SimpleMatrix<T>& other) {
    assert(entity.size() == other.entity.size() && ecols == other.ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] += other.entity[i];
    return *this;
  }
  inline       SimpleMatrix<T>  operator -  (const SimpleMatrix<T>& other) const {
    auto res(*this);
    return res -= other;
  }
  inline       SimpleMatrix<T>& operator -= (const SimpleMatrix<T>& other) {
    return *this += - other;
  }
  inline       SimpleMatrix<T>  operator *  (const T& other) const {
    auto res(*this);
    return res *= other;
  }
  inline       SimpleMatrix<T>& operator *= (const T& other) {
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] *= other;
    return *this;
  }
  inline       SimpleMatrix<T>  operator *  (const SimpleMatrix<T>& other) const {
    assert(ecols == other.entity.size() && entity.size() && other.entity.size());
    auto            derived(other.transpose());
    SimpleMatrix<T> res(entity.size(), other.ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++) {
            SimpleVector<T>& resi(res.entity[i]);
      const SimpleVector<T>& ei(entity[i]);
      for(int j = 0; j < other.ecols; j ++)
        resi[j] = ei.dot(derived.entity[j]);
    }
    return res;
  }
  inline       SimpleMatrix<T>& operator *= (const SimpleMatrix<T>& other) {
    return *this = *this * other;
  }
  inline       SimpleVector<T>  operator *  (const SimpleVector<T>& other) const {
    assert(ecols == other.size());
    SimpleVector<T> res(entity.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      res[i] = entity[i].dot(other);
    return res;
  }
  inline       SimpleMatrix<T>  operator /  (const T& other) const {
    auto res(*this);
    return res /= other;
  }
  inline       SimpleMatrix<T>& operator /= (const T& other) {
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i] /= other;
    return *this;
  }
  inline       SimpleMatrix<T>& operator =  (const SimpleMatrix<T>& other) {
    ecols  = other.ecols;
    entity = other.entity;
    return *this;
  }
  inline       SimpleMatrix<T>& operator =  (SimpleMatrix<T>&& other) {
    ecols  = move(other.ecols);
    entity = move(other.entity);
    return *this;
  }
  inline       bool             operator == (const SimpleMatrix<T>& other) const {
    return ! (*this != other);
  }
  inline       bool             operator != (const SimpleMatrix<T>& other) const {
    if(entity.size() != other.entity.size() || ecols != other.ecols)
      return true;
    for(int i = 0; i < entity.size(); i ++)
      if(entity[i] != other.entity[i]) return true;
    return false;
  }
  inline       T&               operator () (const int& y, const int& x) {
    assert(0 <= y && y < entity.size());
    return entity[y][x];
  }
  inline const T&               operator () (const int& y, const int& x) const {
    assert(0 <= y && y < entity.size());
    return entity[y][x];
  }
  inline       SimpleVector<T>& row(const int& y) {
    assert(0 <= y && y < entity.size());
    return entity[y];
  }
  inline const SimpleVector<T>& row(const int& y) const {
    assert(0 <= y && y < entity.size());
    return entity[y];
  }
  inline const SimpleVector<T>  col(const int& x) const {
    assert(0 <= entity.size() && 0 <= x && x < ecols);
    SimpleVector<T> res(entity.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      res[i] = entity[i][x];
    return res;
  }
  inline       void             setCol(const int& x, const SimpleVector<T>& other) {
    assert(0 <= x && x < ecols && other.size() == entity.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i][x] = other[i];
    return;
  }
  // N.B. transpose : exhaust of the resource, so Eigen library handles better.
  inline       SimpleMatrix<T>  transpose() const {
    SimpleMatrix<T> res(ecols, entity.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < ecols; i ++) {
      SimpleVector<T>& resi(res.entity[i]);
      for(int j = 0; j < entity.size(); j ++)
        resi[j] = entity[j][i];
    }
    return res;
  }
  inline       SimpleMatrix<T>  subMatrix(const int& y, const int& x, const int& h, const int& w) const {
    assert(0 <= h && 0 <= w && 0 <= y && y + h <= rows() && 0 <= x && x + w <= cols());
    SimpleMatrix<T> res(h, w);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = y; i < y + h; i ++)
      for(int j = x; j < x + w; j ++)
        res(i - y, j - x) = (*this)(i, j);
    return res;
  }
  inline       SimpleMatrix<T>& setMatrix(const int& y, const int& x, const SimpleMatrix<T>& d) {
    assert(0 <= y && y + d.rows() <= rows() && 0 <= x && x + d.cols() <= cols());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = y; i < y + d.rows(); i ++)
      for(int j = x; j < x + d.cols(); j ++)
        (*this)(i, j) = d(i - y, j - x);
    return *this;
  }
  inline       SimpleMatrix<T>& O(const T& r = T(int(0))) {
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < rows(); i ++)
      for(int j = 0; j < cols(); j ++)
        (*this)(i, j) = r;
    return *this;
  }
  inline       SimpleMatrix<T>& I(const T& r = T(int(1))) {
    const static T zero(0);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < rows(); i ++)
      for(int j = 0; j < cols(); j ++)
        (*this)(i, j) = (i == j ? r : zero);
    return *this;
  }
  inline       T                determinant(const bool& nonzero = false) const;
  inline       SimpleMatrix<T>  inverse() const {
    // XXX: extremely slow implementation.
    SimpleMatrix<T> result(entity.size(), ecols);
    result.I();
    for(int i = 0; i < result.cols(); i ++)
      result.setCol(i, solve(result.col(i)));
    return result;
  }
  inline       SimpleVector<T>  solve(SimpleVector<T> other) const;
  inline       SimpleVector<T>  solveN(SimpleVector<T> other) const;
  inline       SimpleVector<T>  projectionPt(const SimpleVector<T>& other) const;
  inline       SimpleMatrix<T>& fillP(const vector<int>& idx);
  inline       SimpleMatrix<T>  QR() const;
  inline       SimpleMatrix<T>  SVDleft1d() const;
  inline       pair<SimpleMatrix<T>, SimpleMatrix<T> > SVD1d() const;
  inline       SimpleMatrix<T> SVD() const;
  inline       pair<pair<SimpleMatrix<T>, SimpleMatrix<T> >, SimpleMatrix<T> > SVD(const SimpleMatrix<T>& src) const;
  inline       SimpleVector<T>  zeroFix(const SimpleMatrix<T>& A, vector<pair<T, int> > fidx);
  inline       SimpleVector<T>  inner(const SimpleVector<T>& bl, const SimpleVector<T>& bu) const;
  template <typename U> inline SimpleMatrix<U> real() const {
    assert(0 < entity.size() && 0 < ecols);
    SimpleMatrix<U> res(entity.size(), ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      for(int j = 0; j < ecols; j ++)
        res(i, j) = U(entity[i][j].real());
    return res;
  }
  template <typename U> inline SimpleMatrix<U> imag() const {
    assert(0 < entity.size() && 0 < ecols);
    SimpleMatrix<U> res(entity.size(), ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      for(int j = 0; j < ecols; j ++)
        res(i, j) = U(entity[i][j].imag());
    return res;
  }
  template <typename U> inline SimpleMatrix<U> cast() const {
    assert(0 < entity.size() && 0 < ecols);
    SimpleMatrix<U> res(entity.size(), ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      for(int j = 0; j < ecols; j ++)
        res(i, j) = U(entity[i][j]);
    return res;
  }
  inline const int rows() const {
    return entity.size();
  }
  inline const int cols() const {
    return ecols;
  }
  inline       void resize(const int& rows, const int& cols) {
    assert(0 <= rows && 0 <= cols);
    ecols = cols;
    entity.resize(rows);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < entity.size(); i ++)
      entity[i].resize(ecols);
    return;
  }
  myfloat      epsilon() const {
#if defined(_FLOAT_BITS_)
    // N.B. conservative.
    static const auto eps(sqrt(myfloat(int(1)) >> myint(_FLOAT_BITS_ - 1)));
    // static const auto eps(myfloat(int(1)) >> myint(_FLOAT_BITS_ - 1));
#else
    // N.B. conservative.
    static const auto eps(sqrt(std::numeric_limits<myfloat>::epsilon()));
    // static const auto eps(std::numeric_limits<myfloat>::epsilon());
#endif
    return eps;
  }

  // friend ostream& operator << (ostream& os, const SimpleVector<T>& v);
  // friend istream& operator >> (istream& os, SimpleVector<T>& v);

  // this isn't better idea for faster calculations.
  vector<SimpleVector<T> > entity;
  int ecols;
};

template <typename T> inline T SimpleMatrix<T>::determinant(const bool& nonzero) const {
  assert(0 <= entity.size() && 0 <= ecols && entity.size() == ecols);
  T det(1);
  auto work(*this);
  for(int i = 0; i < entity.size(); i ++) {
    int xchg = i;
    for(int j = i + 1; j < entity.size(); j ++)
      if(abs(work.entity[j][i]) > abs(work.entity[xchg][i]))
        xchg = j;
    swap(work.entity[i], work.entity[xchg]);
    const auto& ei(work.entity[i]);
    const auto& eii(ei[i]);
    if(! nonzero || ! i || pow(abs(det), T(int(1)) / T(int(i))) * epsilon() <= abs(eii))
      det *= eii;
    if(ei.dot(ei) * epsilon() < eii * eii) {
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
      for(int j = i + 1; j < entity.size(); j ++) {
        const auto ratio(work.entity[j][i] / eii);
        work.entity[j] -= ei * ratio;
      }
    }
  }
  return det;
}

template <typename T> inline SimpleVector<T> SimpleMatrix<T>::solve(SimpleVector<T> other) const {
  if(! (0 <= entity.size() && 0 <= ecols && entity.size() == ecols && entity.size() == other.size()) ) throw "SimpleMatrix<T>::Solve error";
  auto work(*this);
  for(int i = 0; i < entity.size(); i ++) {
    int xchg = i;
    for(int j = i + 1; j < entity.size(); j ++)
      if(abs(work.entity[j][i]) > abs(work.entity[xchg][i]))
        xchg = j;
    swap(work.entity[i], work.entity[xchg]);
    swap(other[i], other[xchg]);
    const auto& ei(work.entity[i]);
    const auto& eii(ei[i]);
    if(ei.dot(ei) * epsilon() < eii * eii) {
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
      for(int j = i + 1; j < entity.size(); j ++) {
        const auto ratio(work.entity[j][i] / eii);
        work.entity[j] -= ei       * ratio;
        other[j]       -= other[i] * ratio;
      }
    }
  }
  for(int i = entity.size() - 1; 0 <= i; i --) {
    if(work.entity[i][i] == T(int(0))) continue;
    const auto buf(other[i] / work.entity[i][i]);
    if(!isfinite(buf) || isnan(buf)) {
      // assert(!isfinite(work.entity[i][i] / other[i]) || isnan(work.entity[i][i] / other[i]));
      continue;
    }
    other[i] = buf;
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int j = i - 1; 0 <= j; j --)
      other[j] -= other[i] * work.entity[j][i];
  }
  return other;
}

template <typename T> inline SimpleVector<T> SimpleMatrix<T>::solveN(SimpleVector<T> other) const {
  if(! (0 <= entity.size() && 0 <= ecols && entity.size() == ecols && entity.size() == other.size()) ) throw "SimpleMatrix<T>::SolveN error";
  // XXX: stub
  assert(0 && "SimpleMatrix<T>::solveN stub");
  SimpleVector<T> res;
  return res;
}

template <typename T> inline SimpleVector<T> SimpleMatrix<T>::projectionPt(const SimpleVector<T>& other) const {
  assert(0 < entity.size() && 0 < ecols && ecols == other.size());
  // also needs class or this->transpose() * (*this) == I assertion is needed.
  SimpleMatrix<T> work(entity.size(), ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < work.rows(); i ++)
    work.row(i) = entity[i] * entity[i].dot(other);
  SimpleVector<T> res(ecols);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < other.size(); i ++) {
    res[i] = T(int(0));
    for(int j = 0; j < entity.size(); j ++)
      res[i] += work(j, i);
  }
  return res;
}

template <typename T> inline SimpleMatrix<T>& SimpleMatrix<T>::fillP(const vector<int>& idx) {
  int ii(0);
  for(int j = 0; j < cols() && ii < idx.size(); j ++) {
    SimpleVector<T> ek(cols());
    ek.ek(j);
    ek -= this->projectionPt(ek);
    const auto n2(ek.dot(ek));
    if(n2 <= epsilon()) continue;
    assert(0 <= idx[ii] && idx[ii] < this->rows());
    this->row(idx[ii ++]) = ek / sqrt(n2);
  }
  assert(idx.size() <= ii);
  return *this;
}

template <typename T> inline SimpleMatrix<T> SimpleMatrix<T>::QR() const {
  const auto norm2(norm2M(*this));
  if(! isfinite(norm2)) return *this;
  SimpleMatrix<T> Q(min(this->rows(), this->cols()), this->rows());
  Q.O();
  vector<int> residue;
  residue.reserve(Q.rows());
  for(int i = 0; i < Q.rows(); i ++) {
    const auto Atrowi(this->col(i));
    const auto work(Atrowi - Q.projectionPt(Atrowi));
    const auto n2(work.dot(work));
    if(n2 <= norm2 * epsilon())
      residue.emplace_back(i);
    else
      Q.row(i) = work / sqrt(n2);
  }
  return Q.fillP(residue);
}

template <typename T> inline SimpleMatrix<T> SimpleMatrix<T>::SVDleft1d() const {
  // N.B. A = QR, (S - lambda I)x = 0 <=> R^t Q^t U = R^-1 Q^t U Lambda
  //        <=> R^t Q^t U Lambda' = R^-1 Q^t U Lambda'^(- 1)
  //      A := R^t, B := Q^t U, C := Lambda'
  //          (A + A^-t)*B*(C + C^-1) - A^-t B C - A B C^-1 = 2 A B C
  //        <=> (A + A^-t) * B * (C + C^-1) = (2I + 2A^-tA^-1) * ABC
  //        <=> B = A^-1 (2I + 2A^-(t+1))^-1 (A + A^-t) * B *
  //                (C + C^-1) * C^-1
  // N.B. since S is symmetric, singular value on SS^t = QRR^tQ^t is
  //      same square root as singular value on R.
  const auto S(*this * transpose());
  const auto Qt(S.QR());
  const auto A((Qt * S).transpose());
  const auto A1t(A * A.transpose());
        auto Left(A.inverse() * (SimpleMatrix<T>(A.rows(), A.cols()).I(T(int(2))) + A1t.inverse() * T(int(2))).inverse() * (A + A.transpose().inverse()));
        auto Right(SimpleMatrix<T>(Left.rows(), Left.cols()).O());
  for(int i = 0; i < Right.rows(); i ++)
    Right(i, i) = A(i, i) + T(int(1));
  Left  /= sqrt(norm2M(Left));
  Right /= sqrt(norm2M(Right));
  // N.B. now we have B = Left * B * Right.
  static const int p(ceil(sqrt(- log(epsilon()))));
  for(int i = 0; i < p; i ++) {
    Left  *= Left;
    Right *= Right;
  }
  return (Left * Qt * /* U * */ Right).QR() * Qt;
}

template <typename T> inline pair<SimpleMatrix<T>, SimpleMatrix<T> > SimpleMatrix<T>::SVD1d() const {
  if(this->rows() < this->cols()) {
    auto R(this->transpose().SVDleft1d().transpose());
    return make_pair(((*this) * R).QR(), move(R));
  }
  auto L(this->SVDleft1d());
  return make_pair(move(L), (L * (*this)).transpose().QR().transpose());
}

// XXX: O(n^4) over all, we need O(n^3) methods they make SVD1d as SVDnd.
template <typename T> inline SimpleMatrix<T> SimpleMatrix<T>::SVD() const {
  auto sym((*this) * this->transpose());
  assert(sym.rows() == sym.cols());
  auto res(sym);
  res.I();
  for(int i = 0; i <= sym.rows() + 1; i ++) {
    auto svd(sym.SVD1d());
    sym = svd.first * sym * svd.second;
    res = svd.first * res;
  }
  return res;
}

template <typename T> inline pair<pair<SimpleMatrix<T>, SimpleMatrix<T> >, SimpleMatrix<T> > SimpleMatrix<T>::SVD(const SimpleMatrix<T>& src) const {
  const auto norm2(max(norm2M(*this), norm2M(src)));
  if(! isfinite(norm2)) return *this;
  // refered from : https://en.wikipedia.org/wiki/Generalized_singular_value_decomposition .
  assert(this->cols() == src.cols());
  SimpleMatrix<T> C(this->rows() + src.rows(), this->cols());
  C.setMatrix(0, 0, *this);
  C.setMatrix(this->rows(), 0, src);
  const auto P(C.SVD());
  const auto Qt(C.transpose().SVD().transpose());
  const auto D(P.first * C * Qt.transpose());
  SimpleMatrix<T> P1(this->rows(), this->cols());
  SimpleMatrix<T> P2(src.rows(), this->cols());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < P1.rows(); i ++)
    P1.row(i) = P.col(i);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < P2.rows(); i ++)
    P2.row(i) = P.col(i + P1.rows());
  auto U1(P1.SVD());
  auto Wt(P1.transpose().SVD().transpose());
  auto U2(Wt * P2.transpose());
  vector<int> fill;
  fill.reserve(U2.rows());
  for(int i = 0; i < U2.rows(); i ++) {
    const auto n2(U2.row(i).dot(U2.row(i)));
    if(n2 <= epsilon())
      fill.emplace_back(i);
    else
      U2.row(i) /= sqrt(n2);
  }
  return make_pair(make_pair(move(U1), move(U2.fillP(fill))), (Wt * D).transpose().QR() * Qt);
}

template <typename T> inline SimpleVector<T> SimpleMatrix<T>::zeroFix(const SimpleMatrix<T>& A, vector<pair<T, int> > fidx) {
  // N.B. we now have |[A -bb] [x t]| <= 1 condition.
  // N.B. there's no difference |[A - bb] [x t]|^2 <= 1 condition in this.
  //      but not with mixed condition.
  const auto R((*this) * A);
  SimpleVector<T> one(this->cols());
  one.I(T(int(1)));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < fidx.size(); i ++) {
    one[fidx[i].second] = - fidx[i].first;
    fidx[i].first = - T(int(1));
  }
  // we now have: Q [R [x t] ] <= {0, 1}^m cond.
  auto Pb(*this);
  const auto on(projectionPt(one));
  fidx.reserve(fidx.size() + this->cols());
  for(int i = 0; i < this->cols(); i ++)
    fidx.emplace_back(make_pair(abs(on[i]), i));
  sort(fidx.begin(), fidx.end());
  // sort by: |<Q^t(1), q_k>|, we subject to minimize each, to do this,
  //   maximize minimum q_k orthogonality.
  for(int i = 0, idx = 0; i < this->rows() - 1 && idx < fidx.size(); idx ++) {
    const auto& iidx(fidx[idx].second);
    const auto  orth(this->col(iidx));
    const auto  n2(orth.dot(orth));
    if(T(int(0)) < fidx[idx].first &&
       fidx[idx].first < sqrt(one.dot(one)) * epsilon()) {
      static bool shown(false);
      if(! shown) {
        shown = true;
        cerr << "zeroFix: invariant should be 0 but there's no information." <<
          endl << "         so we choice the invariant one loop before." << endl;
      }
      break;
    }
    // N.B. rank(*this) on call is max rank normally, should not be singular.
    //      however, masp calls with rank isn't max cases.
    if(n2 <= epsilon())
      continue;
    Pb = *this;
    // N.B. O(mn) can be writed into O(lg m + lg n) in many core cond.
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int j = 0; j < this->cols(); j ++)
      this->setCol(j, this->col(j) - orth * this->col(j).dot(orth) / n2);
    if(T(int(0)) < fidx[idx].first) {
      const auto rfidxsz(fidx.size());
      fidx.resize(0);
      fidx.reserve(this->cols());
      const auto on(projectionPt(one));
      for(int j = 0; j < this->cols(); j ++)
        fidx.emplace_back(make_pair(abs(on[j]), j));
      sort(fidx.begin(), fidx.end());
      i -= rfidxsz - fidx.size();
    }
    i ++;
  }
  {
    const auto on(projectionPt(one));
    if(sqrt(on.dot(on)) < sqrt(one.dot(one)) * epsilon()) *this = move(Pb);
  }
  // N.B. now we have fix indexes that to be P R [x 1] * t == 0.
  return R.solve((*this) * one);
}

template <typename T> inline SimpleVector<T> SimpleMatrix<T>::inner(const SimpleVector<T>& bl, const SimpleVector<T>& bu) const {
  assert(this->rows() == bl.size() && this->rows() == bu.size() &&
         0 < this->cols() && 0 < this->rows() && this->cols() < this->rows());
  // |(2 / bu) A x - 1 - bl / bu| <= |1 - bl / bu|
  // <=> with (-A, -bu, -bl), |bl| <= |bu|, |(2 / bu) A x - 2| <= 2(1 - bl / bu)
  auto bU(bu);
  auto bL(bl);
  SimpleMatrix<T> A(*this);
  vector<pair<T, int> > fidx;
  for(int i = 0; i < bU.size(); i ++) {
    if(abs(bu[i]) < abs(bl[i])) {
      bU[i] = - bl[i];
      bL[i] = - bu[i];
      A.row(i) = - this->row(i);
    } else if(bu[i] == bl[i])
      fidx.emplace_back(make_pair(- T(int(bu[i] == T(0) ? 0 : 1)), i));
    assert(bL[i] <= bU[i] && abs(bL[i]) <= abs(bU[i]));
    A.row(i) /= (T(2) * bU[i] - bL[i]) / T(2);
    assert(isfinite(A.row(i).dot(A.row(i))));
  }
  // N.B. in zeroFix, we get linear Invariant s.t. |Ax| <= 1 possible enough.
        auto res(A.QR().zeroFix(A, fidx));
  const auto z(*this * res * T(int(8)));
        T    t(int(1));
  for(int i = 0; i < z.size(); i ++)
    if(bu[i] * z[i] < T(int(0))) // N.B.: infeasible.
      continue;
    else if(z[i] != T(int(0)))
      t = bl[i] * z[i] < T(int(0)) ? min(t, bu[i] / z[i]) :
                  min(t, min(bu[i] / z[i], bl[i] / z[i]));
  return res *= t;
}

template <typename T> ostream& operator << (ostream& os, const SimpleMatrix<T>& v) {
  SimpleMatrix<string> buf(v.rows(), v.cols());
  int M(0);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < v.rows(); i ++)
    for(int j = 0; j < v.cols(); j ++) {
      stringstream ss;
      ss << v(i, j);
      buf(i, j) = ss.str();
#if defined(_OPENMP)
#pragma omp critical
#endif
      {
        M = max(int(buf(i, j).size()), M);
      }
    }
  os << "(" << buf.rows() << ", " << buf.cols() << ")" << "[" << endl;
  for(int i = 0; i < buf.rows(); i ++) {
    os << "[";
    for(int j = 0; j < buf.cols(); j ++) {
      for(int k = buf(i, j).size(); k <= M; k ++)
        os << " ";
      os << buf(i, j);
      if(j < buf.cols() - 1) os << ", ";
    }
    os << "]";
    if(i < buf.rows() - 1) os << ", ";
    os << endl;
  }
  os << "]" << endl;
  return os;
}

template <typename T> istream& operator >> (istream& is, SimpleMatrix<T>& v) {
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == '\n') continue;
    else if(c == '(') break;
    is.unget();
    break;
  }
  int r, c;
  is >> r;
  if(r <= 0) return is;
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == '\n') continue;
    else if(c == ',') break;
    is.unget();
    break;
  }
  is >> c;
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == '\n') continue;
    else if(c == ')') break;
    is.unget();
    break;
  }
  if(c <= 0) return is;
  v.resize(r, c);
  int i(0);
  int j(0);
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c == '\n') continue;
    else if(c == '[') break;
    is.unget();
    break;
  }
  for( ; i < v.rows() && ! is.eof() && ! is.bad(); i ++) {
    while(! is.eof() && ! is.bad()) {
      const auto c(is.get());
      if(c == ' ' || c == '\t' || c == '\n') continue;
      else if(c == '[') break;
      is.unget();
      break;
    }
    for(j = 0; j < v.cols() && ! is.eof() && ! is.bad(); j ++) {
      is >> v(i, j);
      if(v.cols() - 1 <= j) {
        j ++;
        break;
      }
      while(! is.eof() && ! is.bad()) {
        const auto c(is.get());
        if(c == ' ' || c == '\t' || c == '\n') continue;
        else if(c == ',') break;
        is.unget();
        break;
      }
    }
    while(! is.eof() && ! is.bad()) {
      const auto c(is.get());
      if(c == ' ' || c == '\t' || c == '\n') continue;
      else if(c == ']') break;
      is.unget();
      break;
    }
    if(v.rows() - 1 <= i) {
      i ++;
      break;
    }
    while(! is.eof() && ! is.bad()) {
      const auto c(is.get());
      if(c == ' ' || c == '\t' || c == '\n') continue;
      else if(c == ',') break;
      is.unget();
      break;
    }
  }
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t' || c =='\n') continue;
    else if(c == ']') break;
    cerr << "XXX SimpleMatrix<T>::operator >> (\']\')" << flush;
    is.unget();
    break;
  }
  while(! is.eof() && ! is.bad()) {
    const auto c(is.get());
    if(c == ' ' || c == '\t') continue;
    else if(c == '\n') break;
    cerr << "XXX SimpleMatrix<T>::operator >> (\'\\n\')" << flush;
    is.unget();
    break;
  }
  if(i < v.rows() || j < v.cols()) {
    cerr << "XXX SimpleMatrix<T>::operator >> (index)" << flush;
    for( ; i < v.rows(); i ++)
      for( ; j < v.cols(); j ++)
        v(i, j) = T(int(0));
  }
  return is;
}

template <typename T> static inline T norm2M(const SimpleMatrix<T>& m) {
  auto norm2(m.row(0).dot(m.row(0)));
  for(int i = 1; i < m.rows(); i ++)
    norm2 = max(norm2, m.row(i).dot(m.row(i)));
  return norm2;
}

template <typename T> static inline SimpleMatrix<T> log(const SimpleMatrix<T>& m) {
  static const int cut(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) * T(int(2)) );
  SimpleMatrix<T> res(m.rows(), m.cols());
  const auto c(sqrt(norm2M(m)) * T(2));
  const auto residue(SimpleMatrix<T>(m.rows(), m.cols()).I() - m / c);
        auto buf(residue);
  res.I(log(c));
  for(int i = 1; 0 < i && i < cut; i ++) {
    res -= buf / T(i);
    buf *= residue;
  }
  return res;
}

template <typename T> static inline SimpleMatrix<T> logSym(const SimpleMatrix<T>& x, const SimpleMatrix<T>& b) {
  // XXX stub.
  assert(x.rows() == x.cols() && b.rows() == b.cols());
  assert(! (x.rows() % b.rows()) && ! (x.cols() % b.cols()));
  for(int i = 0; i < x.rows(); i ++)
    for(int j = i + 1; j < x.cols(); j ++)
      assert(x(i, j) == x(j, i));
  for(int i = 0; i < b.rows(); i ++)
    for(int j = i + 1; j < b.cols(); j ++)
      assert(b(i, j) == b(j, i));
  // N.B. Ux Lx Uxt == X := B^A == Ub Ua exp(La) Lb_k Uat Ubt.
  const auto Ub(b.SVD());
  const auto Ubt(b.transpose().SVD());
  const auto Lb(Ub * b * Ubt.transpose());
        auto Ux(x.SVD());
        auto Uxt(x.transpose().SVD());
  const auto Lx(Ux * x * Uxt.transpose());
  // N.B. Lx == [[La log(Lb_k)]]
  SimpleVector<T> Lawork(Ux.rows());
  Lawork.O();
  for(int i = 0; i < Lawork.size(); i ++)
    // XXX:
    Lawork[i] = log(Lx(i, i) / Lb(i / (x.rows() / b.rows())));
  // N.B. Lawork.subVector... == another subVector in Ua diag(La) Uat condition.
  // XXX: might be a wrong method.
  SimpleMatrix<T> UUb(x.rows(), x.cols());
  auto UUbt(UUb);
  UUb.O();
  UUbt.O();
  for(int i = 0; i < Ub.rows(); i ++)
    for(int j = 0; j < Ub.cols(); j ++) {
      UUb.setMatrix( i * (x.rows() / Ub.rows()),  j * (x.cols() / Ub.cols()),
        SimpleMatrix<T>(Ub.rows(), Ub.cols()).I(Ub(i, j)) );
      UUbt.setMatrix(i * (x.rows() / Ubt.rows()), j * (x.cols() / Ubt.cols()),
        SimpleMatrix<T>(Ubt.rows(), Ubt.cols()).I(Ubt(i, j)) );
    }
  Ux  = Ub.inverse()  * Ux;
  Uxt = Ubt.inverse() * Uxt;
  // N.B. Ux == Ua sqrt(diag(Lawork)) and same on right side.
  // XXX:
    assert(0 && "log (symmetric matrix A, symmetric matrix B) : stub");
  return x;
}

template <typename T> static inline SimpleMatrix<T> exp01(const SimpleMatrix<T>& m) {
  SimpleMatrix<T> res(m.rows(), m.cols());
  static const int cut(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) * T(int(2)) );
  auto buf(m);
  res.I();
  for(int i = 1; 0 < i && i < cut; i ++) {
    res += buf;
    buf *= m / T(i + 1);
  }
  return res;
}

template <typename T> static inline SimpleMatrix<T> exp(const SimpleMatrix<T>& m) {
  const auto p0(ceil(sqrt(norm2M(m))));
#if defined(_FLOAT_BITS_)
  // XXX:
  // myuint p(p0.operator myint());
  myuint p(p0);
#else
  myuint p(p0);
#endif
  if(T(myint(int(1))) < abs(p0 - T(myint(p)))) throw "too large abs num in exp matrix";
  auto mm(exp01(m / T(myint(p))));
  auto res(m);
  for(res.I(); p; mm *= mm, p >>= 1)
    if(bool(p & myuint(myint(int(1)))))
      res *= mm;
  return res;
}

template <typename T> static inline SimpleMatrix<T> pow(const SimpleMatrix<T>& m, const T& p) {
  return exp(log(m) * p);
}

template <typename T> static inline SimpleMatrix<T> powSym(const SimpleMatrix<T>& m, const SimpleMatrix<T>& p) {
  assert(m.rows() == m.cols() && p.rows() == p.cols());
  for(int i = 0; i < m.rows(); i ++)
    for(int j = i + 1; j < m.cols(); j ++)
      assert(m(i, j) == m(j, i));
  for(int i = 0; i < p.rows(); i ++)
    for(int j = i + 1; j < p.cols(); j ++)
      assert(p(i, j) == p(j, i));
  // m = Um Lm Um^t, m^p == Um Lm^p Um^t.
  // p = Up Lp Up^t, exp(log(Lm_i))^p == Up (Lp*log(Lm_i)) Up^t.
  const auto Um(m.SVD());
  const auto Umt(m.transpose().SVD());
  const auto Lm(Um * m * Umt.transpose());
  const auto Up(p.SVD());
  const auto Upt(p.transpose().SVD());
  const auto Lp(Up * p * Upt.transpose());
  SimpleMatrix<T> res(m.rows() * p.rows(), m.cols() * p.cols());
  res.O();
  for(int i = 0; i < m.rows(); i ++) {
    auto work(Up);
    for(int j = 0; j < p.rows(); j ++)
      work(i * p.rows() + j, j) = exp(Lp(j, j)) * Lm(i, i);
    res.setMatrix(i * p.rows(), i * p.cols(), Up * work * Upt);
  }
  // XXX: here might not be trace trustworthy path,
  //      so this might be a wrong method.
  SimpleMatrix<T> UUm(m.rows() * p.rows(), m.cols() * p.cols());
  auto UUmt(UUm);
  UUm.O();
  UUmt.O();
  for(int i = 0; i < m.rows(); i ++)
    for(int j = 0; j < m.rows(); j ++) {
      UUm.setMatrix(i * p.rows(), j * p.cols(),
        SimpleMatrix<T>(m.rows(), m.cols()).I(Um(i, j)));
      UUmt.setMatrix(i * p.rows(), j * p.cols(),
        SimpleMatrix<T>(m.rows(), m.cols()).I(Umt(i, j)));
    }
  return UUm * res * UUmt;
}

template <typename T> SimpleMatrix<complex<T> > dft(const int& size0) {
  const auto size(abs(size0));
  if(! size) {
    const static SimpleMatrix<complex<T> > m0;
    return m0;
  }
  SimpleMatrix<complex<T> > edft( size, size);
  SimpleMatrix<complex<T> > eidft(size, size);
  const auto file(string("./.cache/lieonn/dft-") + to_string(size) +
#if defined(_FLOAT_BITS_)
    string("-") + to_string(_FLOAT_BITS_)
#else
    string("-ld")
#endif
  );
  ifstream cache(file.c_str());
  if(cache.is_open()) {
    cache >> edft;
    cache >> eidft;
    cache.close();
  } else {
    static const auto Pi(T(int(4)) * atan2(T(int(1)), T(int(1))));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < edft.rows(); i ++) {
      for(int j = 0; j < edft.cols(); j ++) {
        const auto theta(- T(int(2)) * Pi * T(i) * T(j) / T(edft.rows()));
        const auto c(cos(theta));
        const auto s(sin(theta));
        edft( i, j) = complex<T>(c,   s);
        eidft(i, j) = complex<T>(c, - s) / complex<T>(T(size));
      }
    }
    ofstream ocache(file.c_str());
    ocache << edft;
    ocache << eidft;
    ocache.close();
  }
  return size0 < 0 ? eidft : edft;
}

template <typename T> SimpleMatrix<T> diff(const int& size0) {
  const auto size(abs(size0));
  if(! size) {
    static const SimpleMatrix<T> m0;
    return m0;
  }
  SimpleMatrix<T> dd;
  SimpleMatrix<T> ii;
  const auto file(string("./.cache/lieonn/diff-") + to_string(size) +
#if defined(_FLOAT_BITS_)
    string("-") + to_string(_FLOAT_BITS_)
#else
    string("-ld")
#endif
  );
  ifstream cache(file.c_str());
  if(cache.is_open()) {
    cache >> dd;
    cache >> ii;
    cache.close();
  } else {
    // N.B. if we return recursive each size diff,
    //      taylor series should be broken.
    auto DD(dft<T>(size));
    auto II(dft<T>(size));
    static const auto Pi(T(int(4)) * atan2(T(int(1)), T(int(1))));
    // N.B. we should start this loop with i == 1 on integrate(diff) or inverse.
    //      we also should start with i == 0 on taylor series.
    //      we select latter one.
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < DD.rows(); i ++)
      DD.row(i) *= complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) / T(DD.rows()));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 1; i < II.rows(); i ++)
      II.row(i) /= complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) / T(DD.rows()));
    // N.B. if we apply DD onto 1 / (1 / f(x)) graph, it's reverse order.
    //      if we average them, it's the only 0 vector.
    // N.B. there exists also completely correct differential matrix,
    //      but it's also be only 0 vector.
    //      (because it's a imaginary part on originals.)
    // N.B. in continuous function, we don't divide dd by &pi;.
    //      (d/dx sum exp(2 Pi i x theta / N) exp(- 2 Pi i y theta / N) f(y)).
    // N.B. from some numerical test, sign on DD, II is reverse side.
    // N.B. In discrete function, we had be chosen ||dd|| := 1, and not now.
    //      (because the (left or right) differential itself cannot be
    //       larger than |x_{k+1}-x_k| < ||x|| in discrete,
    //       sum_0^1 - 2 Pi i (theta/n)^2/2 -&gt; Pi)
    // N.B. if we make plain differential with no error on cosine curve,
    //      it causes constant 0 vector.
    // N.B. if we don't take this real operator, we cannot get better accuracy
    //      result on taylor series.
    dd =   (dft<T>(- size) * DD).template real<T>();
    ii = - (dft<T>(- size) * II).template real<T>();
    ofstream ocache(file.c_str());
    ocache << dd;
    ocache << ii;
    ocache.close();
    cerr << "." << flush;
  }
  return size0 < 0 ? ii : dd;
}

template <typename T> SimpleMatrix<T> diffRecur(const int& size0) {
  const auto size(abs(size0));
  if(! size) {
    static const SimpleMatrix<T> m0;
    return m0;
  }
  SimpleMatrix<T> dd;
  SimpleMatrix<T> ii;
  const auto file(string("./.cache/lieonn/diffrecur-") + to_string(size) +
#if defined(_FLOAT_BITS_)
    string("-") + to_string(_FLOAT_BITS_)
#else
    string("-ld")
#endif
  );
  ifstream cache(file.c_str());
  if(cache.is_open()) {
    cache >> dd;
    cache >> ii;
    cache.close();
  } else {
    dd = SimpleMatrix<T>(size, size).O().setMatrix(0, 0,
         diffRecur<T>(size - 1));
    ii = SimpleMatrix<T>(size, size).O().setMatrix(0, 0,
         diffRecur<T>(- size + 1));
    cerr << "." << flush;
    if(3 < size) {
      dd += diff<T>(  size) +
        SimpleMatrix<T>(size, size).O().setMatrix(1, 1,
          diffRecur<T>(size - 1));
      ii += diff<T>(- size) +
        SimpleMatrix<T>(size, size).O().setMatrix(1, 1,
          diffRecur<T>(- size + 1));
      dd.row(0) /= T(int(2));
      dd.row(dd.rows() - 1) /= T(int(2));
      ii.row(0) /= T(int(2));
      ii.row(dd.rows() - 1) /= T(int(2));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
      for(int i = 1; i < dd.rows() - 1; i ++)
        dd.row(i) /= T(int(3));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
      for(int i = 1; i < ii.rows() - 1; i ++)
        ii.row(i) /= T(int(3));
      ofstream ocache(file.c_str());
      ocache << dd;
      ocache << ii;
      ocache.close();
    }
  }
  return size0 < 0 ? ii : dd;
}

template <typename T> static inline SimpleVector<complex<T> > taylorc(const int& size, const T& step, const T& stepw) {
  const int  step00(max(int(0), min(size - 1, int(floor(step)))));
  const auto residue0(step - T(step00));
  const auto step0(step00 == size - 1 || abs(residue0) <= T(int(1)) / T(int(2)) ? step00 : step00 + 1);
  const auto residue(step - T(step0));
  if(residue == T(int(0))) return SimpleVector<complex<T> >(size).ek(step0);
  const auto residuem(residue - (step - stepw));
  // N.B. following code is equivalent to exp each dft.
  //      this improves both accuracy and speed.
  // N.B. We don't need to matter which sign dft/idft uses till the sign
  //      we multiply is bonded to the transformation.
  static const auto Pi(T(int(4)) * atan2(T(int(1)), T(int(1)) ));
  auto res(dft<T>(- size).row(step0));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < res.size(); i ++)
    res[i] *= step != stepw ? 
      (i ? exp(complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) * residue / T(res.size()) ))
          / complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) / T(res.size()) )
      - exp(complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) * residuem / T(res.size()) ))
          / complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) / T(res.size()) ) :
        complex<T>(T(int(0))) )
      : exp(complex<T>(T(int(0)), - T(int(2)) * Pi * T(i) * residue / T(res.size()) ));
  return dft<T>(size).transpose() * res;
}

template <typename T> static inline SimpleVector<T> taylor(const int& size, const T& step, const T& stepw) {
  return taylorc<T>(size, step, stepw).template real<T>();
}

template <typename T> static inline SimpleVector<T> taylor(const int& size, const T& step) {
  return taylor<T>(size, step, step);
}

template <typename T> static inline SimpleVector<T> linearInvariant(const SimpleMatrix<T>& in) {
  vector<pair<T, int> > sute;
  return in.QR().zeroFix(in, sute);
}

template <typename T> class SimpleSparseVector {
public:
  inline SimpleSparseVector() { ; }
  inline SimpleSparseVector(const int& sute) { assert(sute == 0); }
  inline SimpleSparseVector(const SimpleSparseVector<T>& other) { *this = other; }
  inline SimpleSparseVector(SimpleSparseVector<T>&& other) { *this = other; }
  inline ~SimpleSparseVector() { ; }
  
  inline SimpleSparseVector<T>  operator -  () const {
    auto res(*this);
    for(auto itr(res.entity.begin()); itr != res.entity.end(); ++ itr)
      itr->second = - itr->second;
    return res;
  }
  inline SimpleSparseVector<T>  operator +  (const SimpleSparseVector<T>& other) const {
    auto res(*this);
    return res += other;
  }
  inline SimpleSparseVector<T>& operator += (const SimpleSparseVector<T>& other) {
    for(auto itr(other.entity.begin()); itr != other.entity.end(); ++ itr) {
      if(itr->second == T(int(0))) continue;
      auto search(entity.lower_bound(itr->first));
      if(search == entity.end() || search->first != itr->first)
        (*this)[itr->first] = itr->second;
      else
        search->second += itr->second;
    }
    return *this;
  }
  inline SimpleSparseVector<T>  operator -  (const SimpleSparseVector<T>& other) const {
    auto res(*this);
    return res -= other;
  }
  inline SimpleSparseVector<T>& operator -= (const SimpleSparseVector<T>& other) {
    return *this += - other;
  }
  template <typename U> inline SimpleSparseVector<T>  operator *  (const U& other) const {
    auto res(*this);
    return res *= other;
  }
  template <typename U> inline SimpleSparseVector<T>& operator *= (const U& other) {
    for(auto itr(entity.begin()); itr != entity.end(); ++ itr)
      itr->second *= other;
    return *this;
  }
  template <typename U> inline SimpleSparseVector<T>  operator /  (const U& other) const {
    auto res(*this);
    return res /= other;
  }
  template <typename U> inline SimpleSparseVector<T>& operator /= (const U& other) {
    for(auto itr(entity.begin()); itr != entity.end(); ++ itr)
      itr->second /= other;
    return *this;
  }
  inline       SimpleSparseVector<T>& operator =  (const SimpleSparseVector<T>& other) {
    entity = other.entity;
    return *this;
  }
  inline       SimpleSparseVector<T>& operator =  (SimpleSparseVector<T>&& other) {
    entity = move(other.entity);
    return *this;
  }
  inline       bool                   operator != (const SimpleSparseVector<T>& other) const {
    for(auto itr(entity.begin()); itr != entity.end(); ++ itr)
      if(itr->second != other[itr->first])
        return true;
    for(auto itr(other.entity.begin()); itr != other.entity.end(); ++ itr)
      if(itr->second != const_cast<const SimpleSparseVector<T>&>(*this)[itr->first])
        return true;
    return false;
  }
  inline       bool                   operator == (const SimpleSparseVector<T>& other) const {
    return ! (*this != other);
  }
  inline       T  dot         (const SimpleSparseVector<T>& other) const {
    T res(0);
    for(auto itr(other.entity.begin()); itr < other.entity.end(); ++ itr) {
      auto search(entity.lower_bound(itr->first));
      if(search != entity.end() && search->first == itr->first)
        res += search->second * itr->second;
    }
    return res;
  }
  inline       T& operator [] (const int& idx) {
    assert(0 <= idx);
    const auto search(entity.lower_bound(idx));
    if(search != entity.end() && search->first == idx)
      return search->second;
    else
      entity[idx] = T(int(0));
    const auto search2(entity.lower_bound(idx));
    assert(search2 != entity.end() && search2->first == idx);
    return search2->second;
  }
  inline const T& operator [] (const int& idx) const {
    assert(0 <= idx);
    if(entity.size()) {
      const auto search(entity.lower_bound(idx));
      if(search != entity.end() && search->first == idx)
        return search->second;
    }
    const static T zero(0);
    return zero;
  }
  inline void     clear() {
    entity = map<int, T>();
    return;
  }
  inline       map<int, T>& iter() {
    return entity;
  }
  inline const map<int, T>& iter() const {
    return entity;
  }
private:
  map<int, T>  entity;
};

template <typename T> using SimpleSparseMatrix = SimpleSparseVector<SimpleSparseVector<T> >;
template <typename T> using SimpleSparseTensor = SimpleSparseVector<SimpleSparseVector<SimpleSparseVector<T> > >;


// N.B. start raw prediction operations.
// N.B. please refer bitsofcotton/randtools.
template <typename T> static inline T makeProgramInvariantPartial(const T& in, const T& ratio, const bool& on01 = false) {
  auto res(on01 ? in :
    ((atan(- in) / atan(T(int(1))) / T(int(2))) + T(int(1))) / T(int(2)) );
  // N.B. better 0 handling, {0, 1} vanished before.
  res += sqrt(SimpleMatrix<T>().epsilon());
  // N.B. CPU float glitch.
  res /= T(int(1)) + sqrt(sqrt(SimpleMatrix<T>().epsilon()));
  assert(T(int(0)) < res && res <= T(int(1)));
  return res *= ratio;
}

template <typename T> static inline pair<SimpleVector<T>, T> makeProgramInvariant(const SimpleVector<T>& in, const T& index = - T(int(1)), const bool& on01 = false) {
  SimpleVector<T> res(in.size() + (T(int(0)) <= index ? 2 : 1));
  res.setVector(0, in);
  res[in.size()] = T(int(1));
  if(T(int(0)) <= index)
    res[in.size() + 1] = T(index);
  T ratio(0);
  for(int i = 0; i < res.size(); i ++)
    ratio += log(res[i] = makeProgramInvariantPartial<T>(res[i],
                            T(int(1)), on01));
  // N.B. x_1 ... x_n == 1.
  // <=> x_1 / (x_1 ... x_n)^(1/n) ... == 1.
  ratio = isfinite(ratio) ? exp(- ratio / T(res.size())) : T(int(1));
  return make_pair(res *= ratio, ratio);
}

template <typename T> static inline T revertProgramInvariant(const pair<T, T>& in, const bool& on01 = false) {
  const auto r0(in.first / in.second);
  const auto r1(T(int(0)) < r0 ? r0 - floor(r0) : ceil(- r0) + r0);
  const auto r(T(int(0)) == r1 ? T(int(1)) : r1);
  return on01 ? r :
      - tan(max(- T(int(1)) + sqrt(SimpleMatrix<T>().epsilon()),
            min(  T(int(1)) - sqrt(SimpleMatrix<T>().epsilon()),
            r * T(int(2)) - T(int(1)) ))
              * atan(T(int(1))) * T(int(2)) );
}

template <typename T> static inline SimpleVector<T> revertProgramInvariant(const pair<SimpleVector<T>, T>& in, const bool& on01 = false) {
  auto res(in.first);
  for(int i = 0; i < in.first.size(); i ++)
    res[i] = revertProgramInvariant<T>(make_pair(in.first[i], in.second), on01);
  return res;
}

template <typename T> class idFeeder {
public:
  inline idFeeder(const int& size = 1) {
    res.resize(size);
    res.O();
    full = false;
    t = 0;
  }
  inline ~idFeeder() { ; }
  inline const SimpleVector<T>& next(const T& in) {
    if(t < res.size())
      res[t] = in;
    else {
      for(int i = 1; i < res.size(); i ++)
        res[i - 1] = move(res[i]);
      res[res.size() - 1] = in;
    }
    if(res.size() <= ++ t) full = true;
    return res;
  }
  SimpleVector<T> res;
  bool full;
private:
  int  t;
};

template <typename T> const T& sgn(const T& x) {
  static const T zero(0);
  static const T one(1);
  static const T mone(- T(int(1)));
  return x != zero ? (zero < x ? one : mone) : zero;
}

template <typename T> class CatG {
public:
  inline CatG() { ; }
  inline CatG(const int& size0, const vector<SimpleVector<T> >& in);
  inline ~CatG() { ; }
  inline T score(const SimpleVector<T>& in) {
    const auto size(cut.size() - 2);
    assert(0 < size);
    return makeProgramInvariant<T>(tayl(size, in.size()) * in).first.dot(cut) - origin;
  }
  const SimpleMatrix<T>& tayl(const int& size, const int& in) {
    static vector<SimpleMatrix<T> > t;
    if(in < t.size()) {
      if(t[in].rows() && t[in].cols())
        return t[in];
    } else
      t.resize(in + 1, SimpleMatrix<T>());
    t[in].resize(size, in);
    for(int i = 0; i < size; i ++)
      t[in].row(i) = taylor<T>(in,
        (T(i) + T(int(1)) / T(int(2))) * T(in) / T(size),
        (T(i) - T(int(1)) / T(int(2))) * T(in) / T(size));
    return t[in];
  }
  SimpleVector<T> cut;
  T   distance;
  T   origin;
};

template <typename T> inline CatG<T>::CatG(const int& size0, const vector<SimpleVector<T> >& in) {
  const auto size(abs(size0));
  SimpleMatrix<T> A(in.size(), size + 1);
  for(int i = 0; i < in.size(); i ++)
    tayl(size, in[i].size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < in.size(); i ++)
    A.row(i) = makeProgramInvariant(tayl(size, in[i].size()) * in[i]).first;
    // N.B. test for linear ones:
    // A.row(i).setVector(0, tayl(size, in[i].size()) * in[i]);
        auto Pt(A.QR());
        auto Ptb(Pt);
  const auto R(Pt * A);
  SimpleVector<T>   one(Pt.cols());
  SimpleVector<int> fix(one.size());
  one.I(T(int(1)));
  fix.I(false);
  for(int n_fixed = 0, idx = 0;
          n_fixed < Pt.rows() - 1;
          n_fixed ++, idx ++) {
    const auto on(Pt.projectionPt(one));
    if(on.dot(on) < one.dot(one) * Pt.epsilon()) {
      Pt = move(Ptb);
      break;
    }
    vector<pair<T, int> > fidx;
    vector<int> pidx;
    fidx.reserve(on.size());
    pidx.resize(on.size(), 0);
    if(0 < size0) {
      for(int i = 0; i < on.size(); i ++)
        fidx.emplace_back(make_pair(abs(on[i]), i));
    } else {
      for(int i = 0; i < on.size(); i ++) {
        T score(0);
        for(int j = 0; j < in.size(); j ++) {
          const auto lscore(abs(on[i] + on[j]));
          if(score == T(int(0)) || lscore < score) {
            score = lscore;
            pidx[i] = j;
          }
        }
        fidx.emplace_back(make_pair(score, i));
      }
    }
    sort(fidx.begin(), fidx.end());
    if(fidx.size() <= idx) break;
    const auto& iidx(fidx[idx].second);
    if(fix[iidx]) continue;
    const auto  orth(Pt.col(iidx));
    const auto  n2(orth.dot(orth));
    if(n2 <= Pt.epsilon()) continue;
    Ptb = Pt;
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int j = 0; j < Pt.cols(); j ++)
      Pt.setCol(j, Pt.col(j) - orth * Pt.col(j).dot(orth) / n2);
    fix[iidx] = true;
    if(size0 < 0) fix[pidx[iidx]] = true;
  }
  auto ptone(Pt * one);
  if(ptone.dot(ptone) <= Pt.epsilon())
    ptone = Ptb * one;
  cut = R.solve(ptone);
  auto cutn(sqrt(cut.dot(cut)));
  if(cutn != T(int(0))) cut /= cutn;
  auto testv((A * cut).entity);
  sort(testv.begin(), testv.end());
  distance = T(int(0));
  for(int i = 1; i < testv.size(); i ++)
    if(distance <= testv[i] - testv[i - 1]) {
      distance =  testv[i] - testv[i - 1];
      origin   = (testv[i] + testv[i - 1]) / T(int(2));
    }
  return;
}

template <typename T> vector<pair<vector<SimpleVector<T> >, vector<int> > > crush(const vector<SimpleVector<T> >& v, const int& cs, const int& count = 0) {
  assert(0 <= count);
  vector<pair<vector<SimpleVector<T> >, vector<int> > > result;
  if(! v.size() || !v[0].size()) return result;
  int t(0);
  result.emplace_back(pair<vector<SimpleVector<T> >, vector<int> >());
  result[0].first.reserve(v.size());
  result[0].second.reserve(v.size());
  for(int i = 0; i < v.size(); i ++) {
    result[0].first.emplace_back(v[i]);
    result[0].second.emplace_back(i);
  }
  vector<pair<T, pair<int, bool> > > sidx;
  sidx.emplace_back(make_pair(T(int(0)), make_pair(0, false)));
  while(sidx.size() < (count ? count : int(sqrt(T(int(v.size())))))) {
    sort(sidx.begin(), sidx.end());
    int iidx(sidx.size() - 1);
    for( ; - 1 <= iidx; iidx --)
      if(iidx < 0 || (! sidx[iidx].second.second &&
        abs(cs) + 1 < result[sidx[iidx].second.first].first.size() ) ) break;
    if(iidx < 0) break;
    const auto& t(sidx[iidx].second.first);
    CatG<T> catg(cs, result[t].first);
    assert(catg.cut.size());
    vector<SimpleVector<T> > left;
    vector<SimpleVector<T> > right;
    vector<int> lidx;
    vector<int> ridx;
    left.reserve(result[t].first.size());
    right.reserve(result[t].first.size());
    lidx.reserve(result[t].first.size());
    ridx.reserve(result[t].first.size());
    for(int i = 0; i < result[t].first.size(); i ++) {
      const auto score(catg.score(result[t].first[i]));
      (score < T(int(0)) ? left : right).emplace_back(move(result[t].first[i]));
      (score < T(int(0)) ? lidx : ridx).emplace_back(move(result[t].second[i]));
    }
    if(left.size() && right.size()) {
      left.reserve(left.size());
      right.reserve(right.size());
      if(left.size() < right.size()) {
        swap(left, right);
        swap(lidx, ridx);
      }
      result[t].first  = move(left);
      result[t].second = move(lidx);
      sidx[iidx].first  = catg.distance;
      sidx[iidx].second = make_pair(t, false);
      result.emplace_back(make_pair(move(right), move(ridx)));
      sidx.emplace_back(make_pair(catg.distance, make_pair(sidx.size(), false)));
    } else {
      result[t].first  = move(left);
      result[t].second = move(lidx);
      result[t].first.reserve(result[t].first.size() + right.size());
      result[t].second.reserve(result[t].second.size() + ridx.size());
      for(int i = 0; i < right.size(); i ++) {
        result[t].first.emplace_back(move(right[i]));
        result[t].second.emplace_back(move(ridx[i]));
      }
      sidx[iidx].first = catg.distance;
      sidx[iidx].second.second = true;
    }
  }
  return result;
}

template <typename T> static inline vector<pair<vector<SimpleVector<T> >, vector<int> > > crush(const vector<SimpleVector<T> >& v) {
  return crush<T>(v, v[0].size());
}

template <typename T> vector<pair<vector<SimpleVector<T> >, vector<int> > > crushWithOrder(const vector<T>& v, const int& cs, const int& count) {
  vector<SimpleVector<T> > work;
  vector<int> edge;
  // N.B. it's O(v.size()^3 * cs^2).
  work.reserve((v.size() * v.size() * v.size() - 19 * v.size() + 30) / 6);
  edge.reserve(v.size() - 1);
  edge.emplace_back(0);
  for(int i = 3; i < v.size(); i ++) {
    SimpleVector<T> buf(i);
    for(int j = 0; j <= v.size() - i; j ++) {
      for(int k = j; k < j + i; k ++)
        buf[k - j] = v[k];
      if(count < 0) {
        vector<T> wbuf;
        wbuf.reserve(buf.size());
        for(int k = 0; k < buf.size(); k ++)
          wbuf.emplace_back(buf[k]);
        sort(wbuf.begin(), wbuf.end());
        for(int k = 0; k < buf.size(); k ++)
          buf[k] = wbuf[k];
      }
      work.emplace_back(buf);
    }
    edge.emplace_back(work.size());
  }
  auto whole_crush(crush<T>(work, cs, count == - 1 ? 0 : abs(count)));
  vector<pair<vector<SimpleVector<T> >, vector<int> > > res;
  res.reserve(whole_crush.size());
  for(int i = 0; i < whole_crush.size(); i ++) {
    vector<int> idx;
    const auto& sec(whole_crush[i].second);
    idx.reserve(sec.size());
    for(int j = 0; j < sec.size(); j ++)
      idx.emplace_back(sec[j] - *lower_bound(edge.begin(), edge.end(), sec[j]));
    res.emplace_back(make_pair(move(whole_crush[i].first), move(idx)));
  }
  return res;
}

template <typename T> static inline vector<pair<vector<SimpleVector<T> >, vector<int> > > crushWithOrder(const vector<T>& v, const int& cs) {
  return crushWithOrder<T>(v, cs, max(int(2), int(sqrt(T(v.size())))));
}

template <typename T> class P012L {
public:
  inline P012L(const int& step = 1, const int& var = 4) {
    assert(1 < var && 0 < step);
    varlen = var;
    this->step = step;
  }
  inline ~P012L() { ; }
  T next(const SimpleVector<T>& in);
private:
  int varlen;
  int step;
};

template <typename T> inline T P012L<T>::next(const SimpleVector<T>& d) {
  static const T zero(int(0));
         auto    M(zero);
  for(int i = 0; i < d.size(); i ++) {
    if(! isfinite(d[i])) return zero;
    M = max(M, abs(d[i]));
  }
  if(M <= zero) return zero;
  vector<SimpleVector<T> > cache;
  cache.reserve(d.size() - varlen + 2);
  for(int i = 0; i < d.size() - varlen - step + 2; i ++) {
    cache.emplace_back(d.subVector(i, varlen));
    cache[cache.size() - 1][varlen - 1] = d[i + varlen + step - 2];
  }
  const auto cat(crush<T>(cache, cache[0].size(), cache.size()));
  SimpleVector<T> work(varlen);
  for(int i = 1; i < work.size(); i ++)
    work[i - 1] = d[i - work.size() + d.size()];
  work[work.size() - 1] = zero;
  auto res(zero);
  auto sscore(zero);
  for(int i = 0; i < cat.size(); i ++) {
    if(! cat[i].first.size()) continue;
    if(! (cat[i].first.size() <= cat[i].first[0].size() + 1)) cerr << "!" << flush;
    SimpleVector<T> avg(cat[i].first[0].size() + 1);
    avg.O();
    for(int j = 0; j < cat[i].first.size(); j ++)
      avg += makeProgramInvariant<T>(cat[i].first[j]).first;
    work[work.size() - 1] = T(int(0));
    const auto avg0(avg);
          auto last(sqrt(work.dot(work)));
    const auto navg(avg.dot(avg));
    if(! isfinite(navg) || navg == zero) continue;
    for(int ii = 0;
            ii < 2 * int(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) )
            && sqrt(work.dot(work) * SimpleMatrix<T>().epsilon()) <
                 abs(work[work.size() - 1] - last); ii ++) {
      last  = work[work.size() - 1];
      const auto vdp(makeProgramInvariant<T>(work));
      avg  = avg0 * sqrt(vdp.first.dot(vdp.first) / avg0.dot(avg0));
      work[work.size() - 1] =
        revertProgramInvariant<T>(make_pair(avg[varlen - 1] /
             T(int(avg.size())), vdp.second));
    }
    const auto vdp(makeProgramInvariant<T>(work));
    T score(0);
    for(int j = 0; j < work.size(); j ++)
      score += work[j] * revertProgramInvariant<T>(make_pair(avg[j], vdp.second));
    res += score * work[work.size() - 1];
    sscore += abs(score * sqrt(work.dot(work)));
  }
  return sscore == zero ? sscore : res / sscore;
}


template <typename T> SimpleVector<T> minsq(const int& size) {
  assert(1 < size);
  const T xsum(size * (size - 1) / 2);
  const T xdot(size * (size - 1) * (2 * size - 1) / 6);
  const auto denom(xdot * T(size) - xsum * xsum);
  SimpleVector<T> s(size);
  for(int i = 0; i < s.size(); i ++)
    s[i] = (T(i) * T(size) - xsum) / denom;
  return s;
}

#if !defined(_PNEXT_ON_MEMORY_)
#define _PNEXT_ON_MEMORY_ true
#endif
template <typename T> const SimpleVector<T>& pnextcacher(const int& size, const int& step) {
  assert(0 < size && 0 <= step);
  if(! _PNEXT_ON_MEMORY_) {
    static SimpleVector<T> nonthreadsafe;
    static int thisstep(0);
    if(nonthreadsafe.size() == size && thisstep == step) return nonthreadsafe;
    thisstep = step;
    return nonthreadsafe = (dft<T>(- size) * (dft<T>(size * 2).subMatrix(0, 0, size, size * 2) * taylorc<T>(size * 2, T(step < 0 ? step * 2 : (size + step) * 2 - 1), T(step < 0 ? step * 2 + 2 : (size + step) * 2 - 3)) )).template real<T>();
  }
  static vector<vector<SimpleVector<T> > > cp;
  if(cp.size() <= size)
    cp.resize(size + 1, vector<SimpleVector<T> >());
  if(cp[size].size() <= step)
    cp[size].resize(step + 1, SimpleVector<T>());
  if(cp[size][step].size()) return cp[size][step];
  return cp[size][step] = (dft<T>(- size) * (dft<T>(size * 2).subMatrix(0, 0, size, size * 2) * taylorc<T>(size * 2, T(step < 0 ? step * 2 : (size + step) * 2 - 1), T(step < 0 ? step * 2 + 2 : (size + step) * 2 - 3)) )).template real<T>();
}
#undef _PNEXT_ON_MEMORY_

template <typename T> const SimpleVector<T>& mscache(const int& size) {
  assert(0 < size);
  static vector<SimpleVector<T> > ms;
  if(ms.size() <= size) ms.resize(size + 1, SimpleVector<T>());
  if(ms[size].size()) return ms[size];
  return ms[size] = minsq<T>(size);
}

template <typename T> const SimpleMatrix<complex<T> >& dftcache(const int& size) {
  assert(size != 0);
  static vector<SimpleMatrix<complex<T> > > cdft;
  static vector<SimpleMatrix<complex<T> > > cidft;
  if(0 < size) {
    if(cdft.size() <= size) cdft.resize(size + 1, SimpleMatrix<complex<T> >());
    if(cdft[size].rows() && cdft[size].cols()) return cdft[size];
    return cdft[size] = dft<T>(size);
  }
  if(cidft.size() <= abs(size)) cidft.resize(abs(size) + 1, SimpleMatrix<complex<T> >());
  if(cidft[abs(size)].rows() && cidft[abs(size)].cols()) return cidft[abs(size)];
  return cidft[abs(size)] = dft<T>(size);
}

template <typename T> class P0 {
public:
  inline P0(const int& step = 1) {
    this->step = step;
  }
  inline ~P0() { ; };
  inline T next(const SimpleVector<T>& in, const int& sute = 0) {
    return pnextcacher<T>(in.size(), ((step - 1) % in.size()) + 1).dot(in);
  }
  int step;
};

template <typename T, typename P> class P0inv {
public:
  inline P0inv() { ; }
  inline P0inv(P&& p) { this->p = p; }
  inline ~P0inv() { ; }
  inline T next(const SimpleVector<T>& in) {
    static const T zero(int(0));
    static const T one(int(1));
    auto ff(in);
    for(int i = 0; i < in.size(); i ++) if(in[i] == zero) return in[in.size() - 1];
    else ff[i] = one / in[i];
    const auto pn(p.next(ff));
    if(pn == zero) return in[in.size() - 1];
    return one / pn;
  }
  P p;
};

template <typename T, typename P> class northPole {
public:
  inline northPole() { ; }
  inline northPole(P&& p) { this->p = p; }
  inline ~northPole() { ; }
  inline T next(const SimpleVector<T>& in, const int& unit = 3) {
    static const T zero(int(0));
    static const T one(int(1));
    static const T M(atan(one / sqrt(SimpleMatrix<T>().epsilon())));
    auto ff(in);
    for(int i = 0; i < in.size(); i ++)
      if(! isfinite(in[i]) || in[i] == zero) return in[in.size() - 1];
      else {
        ff[i] = atan(in[i]);
        // assert(- M < ff[i] && ff[i] < M);
        // N.B. we don't avoid right hand side, it's harmless.
        // ff[i] = atan(one / ff[i]);
        // assert(- M < ff[i] && ff[i] < M);
      }
    auto work(p.next(ff, unit));
    // if(! isfinite(work) || work == zero) return in[in.size() - 1];
    if(! isfinite(work)) return in[in.size() - 1];
    // work = tan(max(- M, min(M, one / tan(max(- M, min(M, work))))));
    work = tan(max(- M, min(M, work)));
    if(isfinite(work)) return work;
    return in[in.size() - 1];
  }
  P p;
};

template <typename T, typename P, bool avg = false> class sumChain {
public:
  inline sumChain() { ; }
  inline sumChain(P&& p) { this->p = p; }
  inline ~sumChain() { ; }
  inline T next(const SimpleVector<T>& in, const int& unit = 3) {
    auto ff(in);
    for(int i = 1; i < ff.size(); i ++)
      ff[i] += ff[i - 1];
    if(! avg) return p.next(ff, unit) - ff[ff.size() - 1];
    const auto A(ff[ff.size() - 1] / T(ff.size()));
    for(int i = 0; i < ff.size(); i ++)
      ff[i] = in[i] - A;
    return p.next(ff, unit) + A;
  }
  P p;
};

template <typename T, typename P> class logChain {
public:
  inline logChain() { ; }
  inline logChain(P&& p) { this->p = p; }
  inline ~logChain() { ; }
  inline T next(const SimpleVector<T>& in, const int& unit = 3) {
    static const T zero(int(0));
    static const T one(int(1));
    auto ff(in);
    if(ff[0] == zero) return in[in.size() - 1];
    for(int i = 1; i < ff.size(); i ++)
      if((ff[i] += ff[i - 1]) == zero) return in[in.size() - 1];
    SimpleVector<T> gg(ff.size() - 1);
    gg.O();
    for(int i = 1; i < ff.size(); i ++)
      if(! isfinite(gg[i - 1] = ff[i] / ff[i - 1] - one)) return in[in.size() - 1];
    return p.next(gg, unit) * ff[ff.size() - 1];
  }
  P p;
};

template <typename T> class P0maxRank0 {
public:
  inline P0maxRank0(const int& step = 1) {
    p = p0_0t(P0<T>(step));
    q = p0_i0t(p0_0t(P0<T>(step)));
  }
  inline ~P0maxRank0() { ; }
  inline T next(const SimpleVector<T>& in, const int& sute = 0) {
    return (p.next(in) + q.next(in)) / T(int(2));
  }
  // N.B. on existing taylor series.
  //      if the sampling frequency is not enough, middle range of the original
  //      function frequency (enough large bands) will effect prediction fail.
  //      this is because we only observes highest and lowest frequency on
  //      sampling points, so omitted part exists.
  //      even if the parameter on P0 is large, situation unchange.
  //      so we should use sectional measurement for them.
  // N.B. the sectional measurament is done by following Ppad class.
  //      So this is only the raw prediction.
  typedef sumChain<T, P0<T>, true> p0_0t;
  typedef P0inv<T, p0_0t> p0_i0t;
  p0_0t p;
  p0_i0t q;
};

template <typename T> class P0maxRank {
public:
  inline P0maxRank(const int& step = 1) {
    p = p0_t(p0_2t(p0_1t(p0_0t(step))));
  }
  inline ~P0maxRank() { ; }
  inline T next(const SimpleVector<T>& in, const int& sute = 0) {
    return p.next(in, sute);
  }
/*
  // N.B. make information-rich not to associative/commutative.
  //      2 dimension semi-order causes (x, status) from input as sedenion.
  // N.B. we need only once P0DFT in general because associative condition
  //      is necessary for input ordering.
  // N.B. we eliminated them. we're better to handle their structures than here.
  // typedef P0DFT<T, p0_1t> p0_2t;
  // N.B. on any R to R into reasonable taylor.
  typedef northPole<T, p0_1t> p0_6t;
  typedef northPole<T, p0_6t> p0_7t;
  // N.B. we treat periodical part as non aligned complex arg part.
  typedef logChain<T, p0_7t>  p0_8t;
  typedef logChain<T, p0_8t>  p0_9t;
  // N.B. we make the prediction on (delta) summation.
  typedef sumChain<T, p0_9t>  p0_10t;
  // N.B. we take average as origin of input.
  typedef sumChain<T, p0_10t, true> p0_t;
  // N.B. this needs huge memory to run.
*/
  // N.B. plain complex form.
  typedef P0maxRank0<T> p0_0t;
  typedef northPole<T, p0_0t>  p0_1t;
/*
  typedef northPole<T, p0_1t> p0_2t;
  typedef logChain<T, p0_2t>  p0_3t;
  typedef logChain<T, p0_3t>  p0_4t;
  typedef sumChain<T, p0_4t>  p0_5t;
  typedef sumChain<T, p0_5t, true> p0_t;
*/
  // N.B. we only handle lebesgue measurable and R(finite)-valued functions.
  //      so worse structures are handled by P01.
  typedef sumChain<T, p0_1t>  p0_2t;
  typedef sumChain<T, p0_2t, true> p0_t;
  p0_t p;
};

template <typename T> class P01delim {
public:
  inline P01delim(const int& step = 1) { ; }
  inline ~P01delim() { ; }
  inline const T& next(const SimpleVector<T>& in, const int& sute = 0) {
    return in[in.size() - 1];
  }
};

// Get invariant structure that
// \[- &alpha, &alpha;\[ register computer with deterministic calculation.
// cf. bitsofcotton/randtools .
// N.B. P = P01 recursive nothing effects in theoretical (f_0(f_1...) == f)
//      however, in the information amount meaning, 6 recursive effects well.
//      (2nd order saturation on the copied information on structure)
//      on some of the experiments, no improves but different result.
template <typename T, typename P = P0maxRank<T>, bool nonlinear = true> class P01 {
public:
  inline P01(const int& step = 1, const int& var = 4) {
    assert(0 < var && 0 < step);
    this->varlen = var;
    this->step = step;
  }
  inline ~P01() { ; }
  inline T next(const SimpleVector<T>& in, const int& unit = 3) {
    static const T zero(0);
    static const T one(1);
    static const T two(2);
    // N.B. please use catgp to compete with over learning.
    // N.B. division accuracy glitch.
    const auto nin(sqrt(in.dot(in) * (one + SimpleMatrix<T>().epsilon())));
    if(! isfinite(nin) || nin == zero) return zero;
    // N.B. if we use last delimiter with unit size, they also causes
    //      many of the exhaust of the calculation resource can be cached
    //      and unstable result, this might means invariant continuity
    //      improves when length == 3 also we have prediction direction on them.
    SimpleMatrix<T> invariants(typeid(P) == typeid(P0maxRank<T>) ? 3
      : (typeid(P) == typeid(P01delim<T>) ? 1 : in.size() - unit), nonlinear ? varlen + 2 : varlen);
    invariants.O();
    for(int i0 = 0; i0 < invariants.rows(); i0 ++) {
      SimpleMatrix<T> toeplitz(in.size() - varlen - step + 2
                               - invariants.rows() + 1, invariants.cols());
      for(int i = i0; i < toeplitz.rows() + i0; i ++) {
        auto work(in.subVector(i, varlen));
        work[work.size() - 1] = in[i + varlen + step - 2];
        toeplitz.row(i - i0) = nonlinear ? makeProgramInvariant<T>(move(work),
          T(i + 1) / T(toeplitz.rows() + 1) ).first : move(work);
      }
      invariants.row(i0) = linearInvariant<T>(toeplitz);
    }
    SimpleVector<T> invariant(invariants.cols());
    invariant.O();
    for(int i = 0; i < invariants.cols(); i ++)
      invariant[i] = typeid(P) == typeid(P01delim<T>)
        ? invariants(invariants.rows() - 1, i)
        : P().next(invariants.col(i), unit);
    if(invariant[varlen - 1] == zero) {
      cerr << "!" << flush;
      return zero;
    }
    SimpleVector<T> work(varlen);
    for(int i = 1; i < work.size(); i ++)
      work[i - 1] = in[i - work.size() + in.size()];
    work[work.size() - 1] = zero;
    if(nonlinear) {
      auto last(sqrt(work.dot(work)));
      for(int ii = 0;
              ii < 2 * int(- log(SimpleMatrix<T>().epsilon()) / log(two) )
              && sqrt(work.dot(work) * SimpleMatrix<T>().epsilon()) <
                   abs(work[work.size() - 1] - last); ii ++) {
        last = work[work.size() - 1];
        const auto work2(makeProgramInvariant<T>(work, one));
        work[work.size() - 1] = revertProgramInvariant<T>(make_pair(
                 - (invariant.dot(work2.first) -
                        invariant[varlen - 1] * work2.first[varlen - 1]) /
                   invariant[varlen - 1], work2.second));
      }
      return work[work.size() - 1];
    }
    return - invariant.dot(work) / invariant[varlen - 1];
  }
private:
  int varlen;
  int step;
};

// N.B. we omit high frequency part (1/f(x) input) to be treated better in P.
template <typename T, typename P> class PBond {
public:
  inline PBond() { ; }
  inline PBond(const int& status, P&& p = P()) {
    assert(0 < status);
    this->p = p;
    f = idFeeder<T>(status);
    M = T(int(1));
  }
  inline ~PBond() { ; }
  inline T next(const T& in) {
    M = max(M, abs(in));
    auto g(f.next(in));
    if(! f.full) return T(int(0));
    // N.B. with 1-norm normalized input:
    T m(g[0] /= M);
    for(int i = 1; i < g.size(); i ++) m = min(m, g[i] /= M);
    // N.B. offset const.
    m -= T(int(1));
    // N.B. 0 < v, normalize with v's orthogonality:
    T mavg(log(g[0] - m));
    for(int i = 1; i < g.size(); i ++) mavg += log(g[i] - m);
    mavg /= T(int(g.size()));
    mavg  = exp(mavg);
    // N.B. we need nonlinear prediction, so * M before to predict.
    return max(- M, min(M, p.next(g / mavg * M) * mavg));
  }
  idFeeder<T> f;
  P p;
  T M;
};

// N.B. if we use each progression for average into input one, they can
//      improve the result. however, if prediction is almost linear,
//      it's useless enough. if the prediction is valid for any input
//      they either useless.
// N.B. if we use some delta into input, they can be progression like effects.
//      however, this makes hypothesis the input is smaller than we get on the
//      stream. this is the condition we lose the interest. so implementation
//      had be eliminated.

// N.B. start det diag operations.
// N.B. invariant gathers some of the group on the input pattern.
template <typename T> SimpleMatrix<T> concat(const SimpleMatrix<T>& m0, const SimpleMatrix<T>& m1) {
  // det diag result = det diag m0 + det diag m1
  // [1 x x^reverse 1]
  // if we met rank shrink, assert exit then.
  // we can handle this with compiling operation adding period-depend values.
  assert(m0.rows() == m1.rows() && m0.cols() == m1.cols());
  SimpleMatrix<T> work0(m0);
  SimpleMatrix<T> work1(m1);
  auto res(m0);
  for(int i = 0; i < m0.rows(); i ++) {
    auto qw1(work1.transpose().QR());
    auto rw1(qw1 * work1.transpose());
    // XXX : assert exit here.
    work0 = (rw1.inverse() * qw1 * work0.transpose()).transpose();
    assert(work0.rows() == work0.cols());
    SimpleMatrix<T> lwork(work0.rows() * 2, work0.cols() * 2);
    const auto ii(SimpleMatrix<T>(work0.rows(), work0.cols()).I());
    lwork.setMatrix(0, 0, ii).setMatrix(0, work0.cols(), work0 - ii).setMatrix(work0.rows(), work0.cols(), ii);
    for(int j = 0; j < work0.rows(); j ++)
      for(int k = 0; k < work0.cols(); k ++)
        lwork(lwork.rows() - j, work0.cols() - k) = work0(j, k);
    for(int j = 0; j < lwork.rows() / 2; j ++)
      for(int k = 0; k < lwork.cols(); k ++)
        swap(lwork(j, k), lwork(lwork.rows() - j, k));
    for(int j = 0; j < lwork.rows(); j ++)
      for(int k = 0; k < lwork.cols() / 2; k ++)
        swap(lwork(j, k), lwork(j, lwork.cols() - k));
    // LDLt:
/*
    auto L();
    auto D();
    auto Linv();
    for(int j = 0; j < L.rows() / 2; j ++)
      for(int k = 0; k < L.cols(); k ++)
        swap(L(j, k), L(L.rows() - j, k));
    for(int j = 0; j < Linv.rows(); j ++)
      for(int k = 0; k < Linv.cols() / 2; k ++)
        swap(Linv(j, k), Linv(j, Linv.cols() - k));
*/
    // factor apply res, work0, work1:
  }
  return res;
}


template <typename T> SimpleMatrix<T> diff(const SimpleMatrix<T>& m, const int& idx) {
  SimpleMatrix<T> res(m.rows() - 1, m.cols());
  res.O();
  for(int i = 0; i < m.rows(); i ++) {
    auto lres(res);
    lres.O();
    for(int j = 0; j < i; j ++) lres.row(j) = m.row(j);
    for(int j = i + 1; j < m.rows(); j ++) lres.row(j - 1) = m.row(j);
    if(m(i, i) == T(int(0))) continue;
    else if(m(i, i) < T(int(0))) lres.row(0) = - lres.row(0);
    res = concat(res, lres /= pow(abs(m(i, i)), T(int(1)) / T(int(lres.rows()))));
  }
}

template <typename T> SimpleMatrix<T> integrate(const SimpleMatrix<T>& m, const int& idx, const int& stage = 0) {
  // N.B. S^x det diag Ax dx (= S u'v) =
  //  (S^x dx) * det diag Ax (= S(uv)') -
  //  S^x(x(det diag Ax)')dx (= S uv')
  //      S^x(x(det diag Ax)')dx   (= S u'v) =
  //  (S^x x dx) * (det diag Ax)'  (= S(uv)') -
  //  S^x(x^2/2 (det diag Ax)'')dx (= S uv')
  //    ...
  SimpleMatrix<T> factorial(m.rows(), m.cols());
  factorial.O();
  for(int i = 0; i < factorial.rows(); i ++)
    factorial(i, idx) = T(int(1)) / T(int(i + 1));
  if(stage == m.rows() - 1) return factorial * m(m.rows() - 1, idx);
  return concat(m, factorial.setMatrix(stage + 1, 0, integrate(diff(m, idx), idx, stage + 1)), true);
}

// N.B. we need huge computing power depends on m at least O(m.rows()^4).
template <typename T> SimpleVector<T> reduce(const SimpleMatrix<T> m) {
  SimpleMatrix<T> work(m);
  for(int i = 0; i < m.rows() - 1; i ++)
    work = integrate(work, i);
  for(int i = 0; i < m.rows() - 1; i ++)
    work = diff(work, i);
  return work.row(0);
}


// N.B. start Decompose
template <typename T> class Decompose {
public:
  typedef SimpleVector<T> Vec;
  typedef SimpleMatrix<T> Mat;
  inline Decompose(const int& size = 0) {
    assert(0 <= size);
    this->size = size;
  }
  inline ~Decompose() { ; }
  const vector<Mat>& A() {
    static vector<vector<Mat> > mA;
    if(mA.size() <= size) mA.resize(size + 1, vector<Mat>());
    auto& a(mA[size]);
    if(a.size() < size) {
      a.reserve(size);
      for(int i = 0; i < size; i ++) {
        SimpleMatrix<T> AA(size, size);
        for(int j = 0; j < AA.rows(); j ++) {
          const auto jj(T(j) * T(i + 2) / T(size + 1));
          AA.row(j) = taylor<T>(AA.cols(), (jj - floor(jj)) * T(size - 1));
        }
        a.emplace_back(move(AA));
      }
      for(int i = 1; i < a.size(); i ++)
        swap(a[a.size() - i], a[a.size() - i - 1]);
    }
    return a;
  }
  inline Vec  mimic(const Vec& dst, const Vec& src, const T& intensity = T(1)) {
    const auto size2(dst.size() / size);
    const auto size3(src.size() / size);
          auto res(dst);
    for(int i = 0; i < size2; i ++) {
      const auto dd(prepare(dst, i));
      apply(res, synth(mother(prepare(src, i * size3 / size2)),
                       freq(mother(dd), dd)) * intensity +
                 dd * (T(1) - intensity), dd, i);
    }
    return res;
  }
  inline Vec  emphasis(const Vec& dst, const T& intensity = T(1)) {
    const auto size2(dst.size() / size);
          auto res(dst);
    for(int i = 0; i < size2; i ++) {
      const auto dd(prepare(dst, i));
            auto lfreq(dd);
      for(int j = 0; j < lfreq.size(); j ++)
        lfreq[j] = T(j) / T(lfreq.size());
      apply(res, synth(mother(dd), lfreq) * intensity +
                 dd * (T(1) - intensity), dd, i);
    }
    return res;
  }
  inline Vec  mother(const Vec& in) {
    vector<Mat> A0;
    if(A0.size() <= size) A0.resize(size + 1, Mat());
    auto& a0(A0[size]);
    if(a0.rows() != size || a0.cols() != size) {
      const auto& a(A());
      a0 = a[0];
      for(int i = 1; i < a.size(); i ++)
        a0 += a[i];
    }
    return a0.solve(in);
  }
  inline Vec  freq(const Vec& mother, const Vec& in) {
    assert(size == mother.size() && size == in.size());
    Mat work(size, size);
    for(int i = 0; i < size; i ++)
      work.setCol(i, A()[i] * mother);
    return work.solve(in);
  }
  inline Vec  synth(const Vec& mother, const Vec& in) {
    assert(size == mother.size() && size == in.size());
    Vec res(size);
    for(int i = 0; i < size; i ++)
      res[i] = T(0);
    for(int i = 0; i < size; i ++)
      res += A()[i] * mother * in[i];
    return res;
  }
  inline Mat subImage(const Mat& img, const int& x, const int& y, const int& r) const {
    Mat res(size, size);
    for(int i = 0; i < res.rows(); i ++)
      for(int j = 0; j < res.cols(); j ++) {
        const auto rr(T(j + 1) / T(res.cols()) * T(r));
        const auto th(T(i) / T(res.rows()) * T(2) * T(4) * atan2(T(1), T(1)));
        res(i, j) = img(flip(x + int(rr * cos(th)), img.rows()),
                        flip(y + int(rr * sin(th)), img.cols()));
      }
    return res;
  }
  Vec  enlarge(const Vec& in, const int& r = 2);
  Mat  represent(const Mat& img, const int& depth = 3);
private:
  inline Vec  prepare(const Vec& in, const int& idx = 0) const {
    const auto cnt(in.size() / size);
    assert(0 < cnt);
    Vec res(size);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < size; i ++)
      res[i] = i * cnt + idx < in.size() ? in[i * cnt + idx] : T(0);
    return res;
  }
  inline void apply(Vec& v, const Vec& dst, const Vec& src, const int& idx = 0) const {
    assert(size && dst.size() == size && src.size() == size);
    const auto cnt(v.size() / size);
    assert(0 < cnt);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(int i = 0; i < size; i ++)
      if(i * cnt + idx < v.size())
        v[i * cnt + idx] += dst[i] - src[i];
    return;
  }
  inline int flip(const int& x, const int& s) const {
    const int xx(abs(x % (s * 2)));
    return s <= xx ? s * 2 - xx - 1 : xx;
  }
  int  size;
};

template <typename T> typename Decompose<T>::Vec Decompose<T>::enlarge(const Vec& in, const int& r) {
  assert(0 < r && size == in.size());
  static vector<vector<Mat> > p;
  if(p.size() <= size)
    p.resize(size + 1, vector<Mat>());
  if(p[size].size() <= r)
    p[size].resize(r + 1, Mat());
  auto& pp(p[size][r]);
  if(pp.rows() < size * r) {
    pp.resize(size * r, size);
    for(int i = 0; i < pp.rows(); i ++)
      pp.row(i) = taylor<T>(size,
        (T(i) + T(int(1)) / T(int(2))) * T(size - 1) / T(pp.rows() - 1),
        (T(i) - T(int(1)) / T(int(2))) * T(size - 1) / T(pp.rows() - 1) );
  }
  const auto m(mother(in));
  const auto f2(freq(m, in));
  const auto bm(pp * m);
        auto ff(bm);
  for(int i = 0; i < ff.size(); i ++)
    ff[i] = i ? f2[(i % (f2.size() - 1)) + 1] : f2[i];
  Decompose<T> ee(size * r);
  auto result(ee.synth(bm, ff));
  return result *= sqrt(in.dot(in) / result.dot(result) * T(r));
}

template <typename T> typename Decompose<T>::Mat Decompose<T>::represent(const Mat& img, const int& depth) {
  Mat res0(1, size);
  Mat w00(img.rows() - size * 2, size);
  const auto int4(diff<T>(- size) * diff<T>(- size) * diff<T>(- size) * diff<T>(- size));
  const auto int4t(int4.transpose());
  for(int i = size; i < img.rows() - size; i ++) {
    Mat w0(img.cols() - size * 2, size);
    for(int j = size; j < img.cols() - size; j ++) {
      vector<Vec> w1;
      for(int r = size;
              r < min(min(i, j),
                    min(img.rows() - i - 1, img.cols() - j - 1));
              r += max(int(1), min(img.rows() / size, img.cols() / size))) {
        // integrate 4th times because we svd 4 times.
        // svd takes bothside transform, we suppose them as differential op.
        const auto part(int4 * subImage(img, i, j, r) * int4t);
        const auto left(part.SVD() * part);
              Vec  work(left.rows());
        for(int k = 0; k < work.size(); k ++)
          work[k] = sqrt(left.row(k).dot(left.row(k))) + T(1);
        // N.B. normalized singular values on the image with circle region.
        //      If this is flat, the data we have is flat.
        //      If this is edged, the data we have has some data.
        work = mother(work);
        // N.B. enlarging specific bias.
        //      recursive on them.
        w1.emplace_back(move(work /= sqrt(work.dot(work))));
      }
      if(! w1.size())
        for(int k = 0; k < w0.cols(); k ++)
          w0(j - size, k) = T(1) / sqrt(T(w0.cols()));
      else if(w1.size() == 1)
        // N.B. line intensity.
        w0.row(j - size) = move(w1[0]);
      else {
        Mat w1m(w1.size(), size);
        for(int i = 0; i < w1m.rows(); i ++)
          w1m.row(i) = move(w1[i]);
        w1m = w1m.transpose();
        const auto left(w1m.SVD() * w1m);
        for(int k = 0; k < left.rows(); k ++)
          w0(j - size, k) = sqrt(left.row(k).dot(left.row(k))) + T(1);
        w0.row(j - size)  = mother(w0.row(j - size));
        w0.row(j - size) /= sqrt(w0.row(j - size).dot(w0.row(j - size)));
      }
    }
    // N.B. do same on x axis: w0 = w0.transpose();
    for(int j = 0; j < w0.rows(); j ++)
      for(int k = 0; k < w0.cols(); k ++)
        assert(isfinite(w0(j, k)) && ! isnan(w0(j, k)));
    const auto left(w0.SVD() * w0);
    for(int k = 0; k < left.rows(); k ++)
      w00(i - size, k) = sqrt(left.row(k).dot(left.row(k))) + T(1);
    w00.row(i - size)  = mother(w00.row(i - size));
    w00.row(i - size) /= sqrt(w00.row(i - size).dot(w00.row(i - size)));
  }
  // N.B. do same on whole image:
  w00 = w00.transpose();
  const auto left(w00.SVD() * w00);
  for(int k = 0; k < left.rows(); k ++)
    res0(0, k) = sqrt(left.row(k).dot(left.row(k))) + T(1);
  res0.row(0)  = mother(res0.row(0));
  res0.row(0) /= sqrt(res0.row(0).dot(res0.row(0)));
  // N.B. recursive on them.
  if(0 < depth && size * 4 <= min(img.rows(), img.cols()) / 2) {
    Mat dimg[5];
    for(int i = 0; i < 5; i ++)
      dimg[i] = Mat(img.rows() / 2, img.cols() / 2);
    for(int i = 0; i < dimg[0].rows(); i ++)
      for(int j = 0; j < dimg[0].cols(); j ++) {
        dimg[0](i, j) = img(i, j);
        dimg[1](i, j) = img(i - dimg[0].rows() + img.rows(), j);
        dimg[2](i, j) = img(i, j - dimg[0].cols() + img.cols());
        dimg[3](i, j) = img(i - dimg[0].rows() + img.rows(),
                            j - dimg[0].cols() + img.cols());
        dimg[4](i, j) = img(i + (img.rows() - dimg[0].rows()) / 2,
                            j + (img.cols() - dimg[0].cols()) / 2);
      }
    Mat dres[5];
    for(int i = 0; i < 5; i ++)
      dres[i] = represent(dimg[i], depth - 1);
    Mat res(1 + dres[0].rows() * 5, size);
    res.row(0) = res0.row(0);
    for(int i = 0; i < 5; i ++)
      for(int j = 0; j < dres[i].rows(); j ++)
        res.row(1 + i * dres[i].rows() + j) = dres[i].row(j);
    return res;
  }
  return res0;
}


// N.B. start image functions
static inline bool whiteline(const string& s) {
  for(auto ss(s.begin()); ss < s.end(); ++ ss)
    if(! std::isspace(* ss) && *ss != '\n')
      return false;
  return true;
}

template <typename T> static inline bool loadstub(istream& input, const int& nmax, const int& ncolor, vector<SimpleMatrix<T> >& datas) {
  int i = 0, j = 0, k = 0;
  char buf;
  int  work = 0;
  bool mode = false;
  while(input.get(buf) && j < datas[0].rows()) {
    if('0' <= buf && buf <= '9') {
      work *= 10;
      work += buf - '0';
      mode  = true;
      continue;
    } else if(mode) {
      mode = false;
      datas[k](j, i) = T(work) / (T(nmax) + T(int(1)));
      work = 0;
      if(++ k >= ncolor) {
        if(++ i >= datas[0].cols()) {
          if(++ j >= datas[0].rows())
            return true;
          i = 0;
        }
        k = 0;
      }
    }
  }
  return true;
}

template <typename T> bool loadp2or3(vector<SimpleMatrix<T> >& data, istream& input) {
  string line;
  string line2;
  string line3;
  try {
    data.resize(3, SimpleMatrix<T>());
    getline(input, line);
    while((whiteline(line) || line[0] == '#') && getline(input, line) && !input.eof() && !input.bad()) ;
    getline(input, line2);
    while((whiteline(line2) || line2[0] == '#') && getline(input, line2) && !input.eof() && !input.bad()) ;
    getline(input, line3);
    while((whiteline(line3) || line3[0] == '#') && getline(input, line3) && !input.eof() && !input.bad()) ;
    istringstream iline2(line2);
    int w, h;
    iline2 >> w;
    iline2 >> h;
    if(line.size() < 2 || w <= 0 || h <= 0) {
      cerr << "unknown size." << endl;
      return false;
    }
    istringstream iline3(line3);
    int nmax;
    iline3 >> nmax;
    if(line[0] == 'P') {
      if(line[1] == '2') {
        data.resize(1);
        data[0] = SimpleMatrix<T>(h,w ).O();
        loadstub<T>(input, nmax, 1, data);
      } else if(line[1] == '3') {
        for(int i = 0; i < 3; i ++)
          data[i] = SimpleMatrix<T>(h, w).O();
        loadstub<T>(input, nmax, 3, data);
      } else {
        cerr << "unknown file type." << endl;
        return false;
      }
    } else {
      cerr << "unknown file type." << endl;
      return false;
    }
  } catch (...) {
    cerr << "Exception while reading." << endl;
    return false;
  }
  return true;
}

template <typename T> bool loadp2or3(vector<SimpleMatrix<T> >& data, const char* filename) {
  ifstream input;
  input.open(filename);
  if(input.is_open()) {
    if(! loadp2or3<T>(data, static_cast<istream&>(input))) {
      input.close();
      return false;
    }
    input.close();
  } else {
    cerr << "Unable to open file for read: " << filename << endl;
    return false;
  }
  return true;
}

template <typename T> bool savep2or3(const char* filename, const vector<SimpleMatrix<T> >& data, const int& depth = 65535) {
  ofstream output;
  output.open(filename);
  if(output.is_open()) {
    try {
      if(data.size() == 1)
        output << "P2" << "\n";
      else
        output << "P3" << "\n";
      output << data[0].cols() << " " << data[0].rows() << "\n";
      output << depth << "\n";
      for(int i = 0; i < data[0].rows(); i ++)
        for(int j = 0; j < data[0].cols(); j ++)
          if(data.size() == 1)
            output << min(int(depth), int(data[0](i, j) * (T(depth) + T(int(1)))) ) << "\n";
          else
            for(int k = 0; k < 3; k ++)
              output << min(int(depth), int(data[k](i, j) * (T(depth) + T(int(1)))) ) << "\n";
    } catch (...) {
      cerr << "An error has occured while writing file." << endl;
    }
    output.close();
  } else {
    cerr << "Unable to open file for write: " << filename << endl;
    return false;
  }
  return true;
}

template <typename T> static inline vector<vector<SimpleMatrix<T> > > normalize(const vector<vector<SimpleMatrix<T> > >& data, const T& upper = T(1)) {
  T MM(0), mm(0);
  bool fixed(false);
  for(int kk = 0; kk < data.size(); kk ++)
    for(int k = 0; k < data[kk].size(); k ++)
      for(int i = 0; i < data[kk][k].rows(); i ++)
        for(int j = 0; j < data[kk][k].cols(); j ++)
          if(! fixed || (isfinite(data[kk][k](i, j)) &&
               ! isinf(data[kk][k](i, j)) && ! isnan(data[kk][k](i, j)))) {
            if(! fixed)
              MM = mm = data[kk][k](i, j);
            else {
              MM = max(MM, data[kk][k](i, j));
              mm = min(mm, data[kk][k](i, j));
            }
            fixed = true;
          }
  auto result(data);
  if(MM == mm) {
    for(int kk = 0; kk < data.size(); kk ++)
      for(int k = 0; k < data[kk].size(); k ++)
        for(int i = 0; i < data[kk][k].rows(); i ++)
          for(int j = 0; j < data[kk][k].cols(); j ++)
            result[kk][k](i, j) = T(int(1)) / T(int(2));
    return result;
  } else if(! fixed)
    return result;
  for(int kk = 0; kk < data.size(); kk ++)
    for(int k = 0; k < data[kk].size(); k ++)
      for(int i = 0; i < data[kk][k].rows(); i ++)
        for(int j = 0; j < data[kk][k].cols(); j ++) {
          if(isfinite(result[kk][k](i, j)) && ! isinf(data[kk][k](i, j)) && ! isnan(result[kk][k](i, j)))
            result[kk][k](i, j) -= mm;
          else
            result[kk][k](i, j)  = T(0);
          assert(T(0) <= result[kk][k](i, j) && result[kk][k](i, j) <= MM - mm);
          result[kk][k](i, j) *= upper / (MM - mm);
        }
  return result;
}

template <typename T> static inline vector<SimpleMatrix<T> > normalize(const vector<SimpleMatrix<T> >& data, const T& upper = T(1)) {
  vector<vector<SimpleMatrix<T> > > w;
  w.emplace_back(data);
  return normalize<T>(w, upper)[0];
}

template <typename T> static inline SimpleMatrix<T> normalize(const SimpleMatrix<T>& in, const T& upper = T(1)) {
  vector<vector<SimpleMatrix<T> > > w;
  w.resize(1);
  w[0].resize(1, in);
  return normalize<T>(w, upper)[0][0];
}

template <typename T> static inline vector<vector<SimpleVector<T> > > normalize(const vector<vector<SimpleVector<T> > >& in, const T& upper = T(1)) {
  vector<vector<SimpleMatrix<T> > > w;
  w.resize(in.size());
  for(int i = 0; i < in.size(); i ++) {
    w[i].resize(in[i].size(), SimpleMatrix<T>(1, in[i][0].size()).O());
    for(int j = 0; j < in[i].size(); j ++)
      w[i][j].row(0) = in[i][j];
  }
  auto res(normalize<T>(w, upper));
  w.resize(0);
  vector<vector<SimpleVector<T> > > v;
  v.resize(res.size());
  for(int i = 0; i < res.size(); i ++) {
    v[i].resize(res[i].size(), SimpleVector<T>(res[i][0].cols()).O());
    for(int j = 0; j < v[i].size(); j ++)
      v[i][j] = res[i][j].row(0);
  }
  return v;
}

template <typename T> static inline SimpleVector<T> normalize(const SimpleVector<T>& in, const T& upper = T(1)) {
  SimpleMatrix<T> w;
  w.resize(1, in.size());
  w.row(0) = in;
  return normalize<T>(w, upper).row(0);
}

template <typename T> static inline vector<SimpleMatrix<T> > autoLevel(const vector<SimpleMatrix<T> >& data, const int& count = 0) {
  vector<T> res;
  res.reserve(data[0].rows() * data[0].cols() * data.size());
  for(int k = 0; k < data.size(); k ++)
    for(int i = 0; i < data[k].rows(); i ++)
      for(int j = 0; j < data[k].cols(); j ++)
        res.emplace_back(data[k](i, j));
  sort(res.begin(), res.end());
  auto result(data);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int k = 0; k < data.size(); k ++)
    for(int i = 0; i < data[k].rows(); i ++)
      for(int j = 0; j < data[k].cols(); j ++)
        result[k](i, j) = max(min(data[k](i, j), res[res.size() - count - 1]), res[count]);
  return result;
}

template <typename T> static inline SimpleVector<T> autoLevel(const SimpleVector<T>& data, const int& count = 0) {
  vector<SimpleMatrix<T> > b;
  b.resize(1);
  b[0].resize(1, data.size());
  b[0].row(0) = data;
  return autoLevel<T>(b, count)[0].row(0);
}

template <typename T> static inline vector<SimpleMatrix<T> > autoGamma(const vector<SimpleMatrix<T> >& data, const T& ratio = T(int(1)) / T(int(2)) ) {
  T r(int(0));
  for(int k = 0; k < data.size(); k ++)
    for(int i = 0; i < data[k].rows(); i ++)
      for(int j = 0; j < data[k].cols(); j ++) {
        assert(T(int(0)) <= data[k](i, j) && data[k](i, j) <= T(int(1)) );
        r += log(data[k](i, j) + T(int(1)) / T(int(65536)) );
      }
  r /= T(int(data.size() * data[0].rows() * data[0].cols()));
  r  = log(ratio) / r;
  auto result(data);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int k = 0; k < data.size(); k ++)
    for(int i = 0; i < data[k].rows(); i ++)
      for(int j = 0; j < data[k].cols(); j ++)
        result[k](i, j) = max(T(int(0)), min(T(int(1)), pow(data[k](i, j) + T(int(1)) / T(int(65536)), r) ));
  return result;
}

template <typename T> static inline SimpleVector<T> autoGamma(const SimpleVector<T>& data, const T& r = T(int(1)) / T(int(2)) ) {
  vector<SimpleMatrix<T> > b;
  b.resize(1);
  b[0].resize(1, data.size());
  b[0].row(0) = data;
  return autoGamma<T>(b, r)[0].row(0);
}

template <typename T> static inline T getImgPt(const T& y, const T& h) {
  auto yy(y % (2 * h));
  if(yy < 0)
    yy = - yy;
  if(yy >= h)
    yy = h - (yy - h);
  return yy % h;
}

template <typename T> SimpleMatrix<T> rotate(const SimpleMatrix<T>& d, const T& 
theta) {
  assert(abs(theta) < atan(T(1)));
  const auto c(cos(theta));
  const auto s(sin(theta));
  const auto h0(abs(int(c * T(d.rows()) - s * T(d.cols()))));
  const auto h1(h0 + abs(int(s * T(d.cols()))) * 2);
  const auto w0(abs(int(s * T(d.rows()) + c * T(d.cols()))));
  const auto w1(w0 + abs(int(s * T(d.rows()))) * 2);
  SimpleMatrix<T> res(h0 < d.rows() ? h1 : h0,
                      w0 < d.cols() ? w1 : w0);
  const T offy(h0 < d.rows() ? abs(int(s * T(d.cols()))) : 0);
  const T offx(w0 < d.cols() ? abs(int(s * T(d.rows()))) : 0);
  res.O();
  const auto diag(int(sqrt(res.rows() * res.rows() +
                           res.cols() * res.cols())) + 1);
  for(int j = - diag; j < diag; j ++)
    for(int k = - diag; k < diag; k ++) {
      const int yy(c * T(j) - s * T(k) + offy);
      const int xx(s * T(j) + c * T(k) + offx);
      if(0 <= yy && yy < res.rows() &&
         0 <= xx && xx < res.cols()) {
        const auto dyy(getImgPt<int>(j, d.rows()));
        const auto dxx(getImgPt<int>(k, d.cols()));
        {
          res(yy, xx) = res(min(yy + 1, int(res.rows()) - 1), xx) =
            res(yy, min(xx + 1, int(res.cols()) - 1)) =
            res(min(yy + 1, int(res.rows()) - 1),
                min(xx + 1, int(res.cols()) - 1)) =
              d(dyy, dxx);
        }
      }
    }
  return res;
}

template <typename T> static inline SimpleMatrix<T> center(const SimpleMatrix<T>& dr, const SimpleMatrix<T>& d) {
  SimpleMatrix<T> res(d.rows(), d.cols());
  for(int i = 0; i < res.rows(); i ++)
    for(int j = 0; j < res.cols(); j ++)
      res(i, j) = dr(max(int(0), min(i + (dr.rows() - d.rows()) / 2, dr.rows() - 1)),
                     max(int(0), min(j + (dr.cols() - d.cols()) / 2, dr.cols() - 1)));
  return res;
}


// N.B. start ddpmopt

// N.B. utility types used in predv.
// N.B. once we had PP3, PP6, PP9 and so on but it's a saturation in
//      second order or more also only a simplicity improves.
//      the result we get isn't depends on simplicity on loop, so we elim it.
// template <typename T> using PP0 = PdeltaOnce<T, P01<T, P0maxRank<T>, true> >;
// N.B. we use maximum of them as invariant compatible, no continuous condition.
// N.B. we make insurance as PdeltaOnce, this causes tan(Ax-x)==tan(Bx),
//      diag(eigen(B)) ~~ diag(1) hypothesis but usually the original stream
//      matrix is larger than this. we might need P0DFT after doing this but
//      they should be included in P01 also friendly to orthogonal matrices.
// N.B. some of the soft PRNG tests doesn't need PdeltaOnce.
// template <typename T> using PP0 = PdeltaOnce<T, P01<T, P01delim<T>, true> >;
// N.B. we are targetting the structure they appears additional states after
//      additional states on given input range. so we don't use PdeltaOnce.
// N.B. once we had P0DFT class around P0 class, they slides DFT coefficients
//      per each and predict with slided ones.
//      handling such a method is much heavier than we now use below
//      and not so efficient than we have. so we eliminated them.
// N.B. once we had the class avoiding timing-related prediction fails with
//      PSVD class. however they hardly depends on phase periods on input
//      we suppose first. so we eliminated them.
// N.B. with using predv with PRNG, we can apply before applying PP0 P012L.
//      our tests on PRNG can improves the result.
template <typename T> using PP0 = P01<T, P01delim<T>, true>;

// N.B. as ddpmopt:README.md, PP3 is least and enough normally.
template <typename T, int nprogress = 20> SimpleVector<T> predv0(const vector<SimpleVector<T> >& in, const string& strloop = string(""), const int& sz = 9, const int& step = 1) {
  assert(0 < sz && sz <= in.size());
  // N.B. we need to initialize p0 vector.
  SimpleVector<T> seconds(sz);
  seconds.O();
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < sz; i ++)  {
    seconds[i] = makeProgramInvariant<T>(in[i], - T(int(1)), true).second;
  }
  // N.B. not in use, we use whole in.size() with PP0.
  const int unit(sz / 3);
  SimpleVector<T> p(in[0].size());
  p.O();
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int j = 0; j < in[0].size(); j ++) {
    if(nprogress && ! (j % max(int(1), int(in[0].size() / nprogress))) )
      cerr << j << " / " << in[0].size() << ", " << strloop << endl;
    idFeeder<T> buf(sz);
    for(int i = 0; i < sz; i ++)
      buf.next(makeProgramInvariantPartial<T>(in[i][j], seconds[i], true));
    assert(buf.full);
    p[j] = PP0<T>(step).next(buf.res, unit);
  }
  const auto nseconds(sqrt(seconds.dot(seconds)));
  return revertProgramInvariant<T>(make_pair(
    makeProgramInvariant<T>(normalize<T>(p), - T(int(1)), true).first,
      PP0<T>(step).next(seconds / nseconds, unit) * nseconds), true);
}

// N.B. as p8:README.md, predv once is enough for finite combinations
//      except for upper cardinals. If we're lucky enough, the original stream
//      is made from Lie algebra with tangent, we reduce combination on them,
//      the residue could be continuous one variable tangent series and
//      we bet them.
//      However, the residue is normally only the noise, so we do once more,
//      also we bet the result to be return to the average or having some
//      of a continuity.
//      We return low prediction we can't bet them raw but is enough to
//      image prediction.
// N.B. predv *quint*
// N.B. over all, we should need around 5 * 60 step input for normal use.
//      this is huge problem for us nor our computer resource.
//
// Important N.B. However, belows doesn't improve output enough, so we elim it.
// N.B. instead of them, we apply P0maxRank0 after predv,
//      this improves well in practical and up to *plain* raw aleph_0.
// N.B. after testing with some of the PRNG series, subtract combination
//      is continuous condition the hypothesis we made -- is continuous gets
//      better results for us.
// N.B. after github.com/bitsofcotton/p2 upload and p1 change,
//      some of the PRNG test meaning broken. so revert them.
//      (changed p1/pp3.cc predv call to predv0 call causes split predictions
//       however the command line chain meaning unchanged.)
// Important N.B. we use this function with delta input, [start + step + 1, -0[
//                summation output.
//                however, we correct them with real input, so raw option isn't
//                needed.
// N.B. we maybe in invariant controlled condition, so return 2 of candidates.
template <typename T, int nprogress = 20> static inline pair<SimpleVector<T>, SimpleVector<T> > predv1(vector<SimpleVector<T> >& in, const int& step = 1) {
  assert(0 < step && 10 + step * 2 <= in.size() && 1 < in[0].size());
  // N.B. we use whole width to get better result in average.
  //      this is equivalent to the command: p1 | p0 :
  //      enough prediction with integer-float classes (SimpleFloat<...>)
  //      takes O(length^2) to get the result in their cases,
  //      however with CPU implemented float they can have some unknown errors,
  //      we should use at least p1-... | p0-... in their case.
  // XXX: If we implement loop hard optimizer, we can use them with
  //      O((length*pixel*accuracy)^2) for each pixel bit even also any of
  //      the contexts they can have implementation.
  SimpleVector<SimpleVector<T> > p;
  SimpleVector<T> res(in[0].size());
  SimpleVector<T> resc(in[0].size());
  p.entity.reserve(in.size());
  // N.B. optimal with PP0
  const auto start(8 + step);
  // N.B. with PSVD we need start *= 3 or we use only 3 blocks.
  // const auto start(max((8 + step) * 3, int(in.size()) - 2));
  for(int i = 1; i < start; i ++)
    p.entity.emplace_back(SimpleVector<T>(in[0].size()).O());
  for(int i = start; i <= in.size(); i ++)
    p.entity.emplace_back(predv0<T, nprogress>(in, to_string(i) + string(" / ") + to_string(in.size()), i, step));
  SimpleMatrix<T> ip(p.size(), res.size());
  for(int i = 0; i < start + step; i ++)
    ip.row(i).O();
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = start + step; i < ip.rows(); i ++) {
    for(int j = 0; j < ip.cols(); j ++)
      ip(i, j) =
        (in[i - ip.rows() + in.size()][j] * T(int(2)) - T(int(1)) ) *
        (p[i - ip.rows() + p.size() - step][j] * T(int(2)) - T(int(1)) );
  }
  // N.B. dftcache need to be single thread on first call.
  // N.B. we bet combination subtracted series is continuous.
  // N.B. once we used differences prediction, but now, we only bet their signs.
  //      so the result is: pred2(in * pred(in)) * pred(in) ~
  //        sgn(pred3(in)) * alpha.
  // N.B. either, diferrences prediction is better friendly to upper layer
  //      monte-carlo methods, however, with some of the test we have isn't
  //      get better result them (only returns last one image), so they're
  //      eliminated.
  // N.B. we're in invariant controlled condition.
  //      so raw real invariant is res, but we return {res, resc} for
  //      res == 0 controlled conditions.
  //      we can use delta input, sum output instead of this,
  //      however we select resc condition because of P01 4 dimension condition.
  //      if we do them with this hack, 4 dimensions we returns.
  resc[0] = res[0] = (P0maxRank0<T>(step).next(ip.col(0)) *
    (p[p.size() - 1][0] * T(int(2)) - T(int(1)) ) + T(int(1)) ) / T(int(2));
  for(int j = start + step + 1; j < p.size() - 1; j ++) {
    cerr << "pnext: " << j << " / " << p.size() - 1 << endl;
    resc[0] -= (P0maxRank0<T>(step).next(ip.col(0).subVector(0, j + 1)) *
      (p[j][0] * T(int(2)) - T(int(1)) ) + T(int(1)) ) / T(int(2));
  }
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 1; i < res.size(); i ++) {
    if(nprogress && ! (i % max(int(1), int(res.size() / nprogress))) )
      cerr << i << " / " << res.size() << endl;
    resc[i] = res[i] = (P0maxRank0<T>(step).next(ip.col(i)) *
      (p[p.size() - 1][i] * T(int(2)) - T(int(1)) ) + T(int(1)) ) / T(int(2));
    for(int j = start + step + 1; j < p.size() - 1; j ++)
      resc[i] -= (P0maxRank0<T>(step).next(ip.col(i).subVector(0, j + 1)) *
        (p[j][i] * T(int(2)) - T(int(1)) ) + T(int(1)) ) / T(int(2));
  }
  in.resize(0);
  return make_pair(res, resc);
}

// N.B. we apply PRNGs before to predict each of prediction in general.
//      this is needed if original streams' entropy feeding is not stable
//      case, the implicit recursive functions' average can be stable with
//      this hack. also we might need P012L before to PP0 on each predv1.
//      this is the condition: non Lebesgure measurable condition can feed
//      each function entropy can be pred success or failure entropy they
//      can slice some amount of prediction bit.
//      however, this method either only slices some amount of the original
//      structure and their internal states by PRNGs.
//      so the output will be obscure enough if we cannot slice original
//      structure enough with ours.
//      in the most of the cases, we don't need P012L with better PRNGs.
//      we suppose phase period doesn't connected to the original structures.
template <typename T, int nrecur = 0, int nprogress = 20> static inline pair<SimpleVector<T>, SimpleVector<T> > predv(vector<SimpleVector<T> >& in, const int& step = 1) {
  if(! nrecur) return predv1<T, nprogress>(in, step);
  pair<SimpleVector<T>, SimpleVector<T> > res;
  res.first.resize(in[0].size());
  res.second.resize(in[0].size());
  res.first.O();
  res.second.O();
  for(int i = 0; i < nrecur; i ++) {
    auto rin(in);
    for(int i = 0; i < rin.size(); i ++)
      for(int j = 0; j < rin[i].size(); j ++)
#if defined(_ARCFOUR_)
        rin[i][j] = (rin[i][j] + T(arc4random_uniform(0x20001)) / T(0x20000)) / T(int(2));
#else
        rin[i][j] = (rin[i][j] + T(random() % 0x20000) / T(0x20000 - 1)) / T(int(2));
#endif
    // N.B. PRNG parts going to gray + small noise with large enough nrecur.
    auto n(predv1<T, nprogress>(rin, step));
    res.first  += n.first;
    res.second += n.second;
  }
  res.first  /= T(nrecur);
  res.second /= T(nrecur);
  return res;
}

// N.B. predv only returns last one picture on some of our tests with real
//      data, but is effective returns better results with PRNG tests.
//      so with real data, we only apply P0maxRank to each pixel.
// N.B. eliminated below. this is because: our test is well done with the
//      3 of combinations candidates normally with PRNGs:
//      {catgp | p1 | p0, p1 | p0, p0} so these are 3 of the measurable
//      conditions we suppose. So we're targetting only stable entropy feed
//      after and after stream, so we make hypothesis Riemann-Stieljes cond.
//      Otherwise, enough continuous datas have Riemann measurable cond.
//      Either, enough complex datas have non Lebesgue measurable cond
//      (No unique pure functions cond nor no non entropy phenomenon cond.).
// N.B. we also eliminated above:
// N.B. instead of recursive doing sliding input length (they hardly depends
//      on first some steps), we use montecarlo method.
//      recur == 11 is enough to get probability around .9 when we're using
//      Condorcet's jury counting.
// N.B. the effect recur doesn't get better ones. So we eliminated them.
// N.B. we eliminated predvall, we don't need them with whole internal states
//      awared predictors they have a better prediction concerned with some
//      series of a PRNG tests.

// N.B. predv4 is for masp generated -4.ppm predictors.
//      mostly with slight speed hacks.
// N.B. we also make hypothesis on this function as delta input,
//      [11, -0[ range summation output.
template <typename T, int nprogress = 6> static inline pair<SimpleVector<T>, SimpleVector<T> > predv4(vector<SimpleVector<T> >& in) {
  assert(1 < in.size() && (in[in.size() - 1].size() == 4 ||
                           in[in.size() - 1].size() == 12) );
  static const T zero(0);
  static const T one(1);
  static const T two(2);
  SimpleVector<T> res(in[in.size() - 2].size());
  SimpleVector<T> resc(in[in.size() - 2].size());
  vector<SimpleVector<T> > inw;
  inw.reserve(in.size());
  SimpleVector<T> nwork(in.size());
  for(int i = 0; i < in.size(); i ++) {
    auto inww(makeProgramInvariant<T>(in[i], - T(int(1)), true));
    inw.emplace_back(inww.first);
    nwork[i] = inww.second;
  }
  SimpleMatrix<T> gwork0(res.size(), inw.size() / 2 - 1);
  gwork0.O();
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < gwork0.rows(); i ++) {
    // N.B. imported from P01 class.
    SimpleMatrix<T> toeplitz0(gwork0.cols(), 7);
    for(int j = 0; j < toeplitz0.rows(); j ++) {
      SimpleVector<T> vw(5);
      vw.O();
      vw.setVector(0, inw[j * 2 + 1].subVector(0, 4));
      // XXX: following throws floating point exception, non zero divide.
      //  1 + ((i / (in[in.size() - 2].size() / in[in.size() - 1].size())) & (~ 0x03)), 4));
      vw[4] = inw[j * 2 + 2][i];
      toeplitz0.row(j) =
        makeProgramInvariant<T>(vw, T(j) / T(int(toeplitz0.rows() + 1)) ).first;
    }
    for(int i1 = 9; i1 <= toeplitz0.rows(); i1 ++) {
      if(nprogress && ! (i1 % max(int(1), int(toeplitz0.rows() / nprogress))) )
        cerr << i1 << " / " << toeplitz0.rows() << " : " << i << " / " << in[0].size() << endl;
      SimpleMatrix<T> toeplitz(i1, toeplitz0.cols());
      for(int j = 0; j < toeplitz.rows(); j ++)
        toeplitz.row(j) = toeplitz0.row(j);
      const auto invariant(linearInvariant<T>(toeplitz));
      SimpleVector<T> work(5);
      work.O();
      work.setVector(0, inw[(i1 - toeplitz0.rows()) * 2 + inw.size() - 1].subVector(0, 4));
      work[work.size() - 1] = zero;
      auto last(sqrt(work.dot(work)));
      for(int ii = 0;
              ii < 2 * int(- log(SimpleMatrix<T>().epsilon()) / log(two) )
              && sqrt(work.dot(work) * SimpleMatrix<T>().epsilon()) <
                 abs(work[work.size() - 1] - last); ii ++) {
        last = work[work.size() - 1];
        const auto work2(makeProgramInvariant<T>(work, one));
        work[work.size() - 1] = revertProgramInvariant<T>(make_pair(
          - (invariant.dot(work2.first) - invariant[4] * work2.first[4]) /
          invariant[4], work2.second));
      }
      gwork0(i, i1 - 1) = work[work.size() - 1];
    }
  }
  for(int i1 = 9; i1 < gwork0.cols(); i1 ++) {
    const auto nnwork(nwork.subVector(0, i1));
    const auto nseconds(sqrt(nnwork.dot(nnwork)));
    gwork0.setCol(i1, revertProgramInvariant<T>(make_pair(
      makeProgramInvariant<T>(normalize<T>(gwork0.col(i1)), - T(int(1)), true).first,
        PP0<T>().next(nnwork / nseconds, 0) * nseconds), true).subVector(0, gwork0.rows()) );
  }
  auto gwork1(gwork0);
  gwork1.O();
  // N.B. imported from predv1.
  for(int i = 0; i < gwork1.rows(); i ++)
    for(int j = 9; j < gwork1.cols(); j ++)
      gwork1(i, j) =
        (in[(j - gwork1.cols()) * 2 + in.size()][i] * T(int(2)) - T(int(1)) ) *
        (gwork0(i, j - 1) * T(int(2)) - T(int(1)) );
  // N.B. same logic as predv, we bet only the sign of them.
  resc[0] = res[0] = (P0maxRank0<T>().next(gwork1.row(0)) *
    (gwork0(0, gwork0.cols() - 1) * T(int(2)) - T(int(1)) ) +
      T(int(1)) ) / T(int(2));
  for(int j = 11; j < gwork1.cols() - 1; j ++) {
    cerr << "pnext: " << j << " / " << gwork1.cols() - 1 << endl;
    resc[0] -= (P0maxRank0<T>().next(gwork1.row(0).subVector(0, j + 1)) *
      (gwork0(0, gwork0.cols() - 1) * T(int(2)) - T(int(1)) ) +
        T(int(1)) ) / T(int(2));
  }
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 1; i < res.size(); i ++) {
    resc[i] = res[i] = (P0maxRank0<T>().next(gwork1.row(i)) *
      (gwork0(i, gwork0.cols() - 1) * T(int(2)) - T(int(1)) ) +
        T(int(1)) ) / T(int(2));
    for(int j = 11; j < gwork1.cols() - 1; j ++)
      resc[i] -= (P0maxRank0<T>().next(gwork1.row(i).subVector(0, j + 1))
        * (gwork0(i, gwork0.cols() - 1) * T(int(2)) - T(int(1)) ) +
          T(int(1)) ) / T(int(2));
  }
  return make_pair(res, resc);
}

template <typename T> pair<vector<SimpleVector<T> >, vector<SimpleVector<T> > > predVec(vector<vector<SimpleVector<T> > >& in0, const int& step = 1) {
  assert(in0.size() && in0[0].size() && in0[0][0].size());
  vector<SimpleVector<T> > in;
  in.resize(in0.size());
  for(int i = 0; i < in0.size(); i ++) {
    assert(in0[i].size() == in0[0].size() &&
           in0[i][0].size() == in0[0][0].size());
    in[i].resize(in0[i].size() * in0[i][0].size());
    for(int j = 0; j < in0[i].size(); j ++) {
      assert(in0[i][0].size() == in0[i][j].size());
      in[i].setVector(j * in0[i][0].size(), in0[i][j]);
    }
  }
  const auto size0(in0[0].size());
  const auto size1(in0[0][0].size());
  in0.resize(0);
  auto p(predv<T>(in, step));
  pair<vector<SimpleVector<T> >, vector<SimpleVector<T> > > res;
  res.first.resize(size0);
  res.second.resize(size0);
  for(int j = 0; j < res.first.size(); j ++) {
    res.first[j]  = p.first.subVector( size1 * j, size1);
    res.second[j] = p.second.subVector(size1 * j, size1);
  }
  return res;
}

// N.B. once we have audo en/decoder which is DFT image[n] DFT with 
//      input prediction direction sliding DFT also doing bit-wise separation
//      giving into predv, however they only returns left-top white point and
//      black outed image or only yellow outed images for a result.
//      so we reverted to original plain predictions.
// N.B. we also did them with result sgn middle stage predv result conversion,
//      however they aren't changed result.
// N.B. original DFT image[n] DFT conversion looks better with continuous
//      input converted into discrete variables, however, our predictor isn't
//      get 100% result, so some probability they slips, so their slips
//      effects whole result graphics in IDFT.
// N.B. sliding prediction direction DFT can have explicit relations to
//      many much more on input graphics, they should works well but not for us.
//      this either can be caused by prediction errors as a predv output.
//      monte-carlo 11*11 times isn't changed this situation.
// N.B. we're now using better algorithm than sliding DFT implemented
//      on predv they uses statistics continuity as half of output.
// N.B. either, linear predictors doesn't affected by such of DFT
//      transformations.
template <typename T> pair<vector<SimpleMatrix<T> >, vector<SimpleMatrix<T> > > predMat(vector<vector<SimpleMatrix<T> > >& in0, const int& step = 1) {
  assert(in0.size() && in0[0].size() && in0[0][0].rows() && in0[0][0].cols());
  vector<SimpleVector<T> > in;
  in.resize(in0.size());
  for(int i = 0; i < in0.size(); i ++) {
    assert(in0[i].size() == in0[0].size());
    in[i].resize(in0[i].size() * in0[i][0].rows() * in0[i][0].cols());
    for(int j = 0; j < in0[i].size(); j ++) {
      assert(in0[i][j].rows() == in0[0][0].rows() &&
             in0[i][j].cols() == in0[0][0].cols());
      for(int k = 0; k < in0[i][j].rows(); k ++)
        in[i].setVector(j * in0[i][0].rows() * in0[i][0].cols() +
                        k * in0[i][0].cols(), in0[i][j].row(k));
    }
  }
  const auto size(in0[0].size());
  const auto rows(in0[0][0].rows());
  const auto cols(in0[0][0].cols());
  in0.resize(0);
  auto p(predv<T>(in, step));
  pair<vector<SimpleMatrix<T> >, vector<SimpleMatrix<T> > > res;
  res.first.resize( size);
  res.second.resize(size);
  for(int j = 0; j < res.first.size(); j ++) {
    res.first[ j].resize(rows, cols);
    res.second[j].resize(rows, cols);
    for(int k = 0; k < rows; k ++) {
      res.first[ j].row(k) = p.first.subVector( j * rows * cols + k * cols, cols);
      res.second[j].row(k) = p.second.subVector(j * rows * cols + k * cols, cols);
    }
  }
  return res;
}

template <typename T> pair<SimpleSparseTensor<T>, SimpleSparseTensor<T> > predSTen(vector<SimpleSparseTensor<T> >& in0, const vector<int>& idx, const int& step = 1) {
  assert(idx.size() && in0.size());
  // N.B. we don't do input scaling.
  // N.B. the data we target is especially string stream corpus.
  //      they are incontinuous one, so complementing with continuous stream
  //      shouldn't improve outputs.
  // N.B. we should use each bit extended input stream but not now.
  //      they uses large enough memory we cannot comput on our machines.
  vector<SimpleVector<T> > in;
  vector<pair<int, pair<int, int> > > attend;
  in.resize(in0.size());
  attend.reserve(idx.size() * idx.size() * idx.size());
  for(int i = 0; i < idx.size(); i ++)
    for(int j = 0; j < idx.size(); j ++)
      for(int k = 0; k < idx.size(); k ++) {
        for(int ii = 0; ii < in0.size(); ii ++)
          if(in0[ii][idx[i]][idx[j]][idx[k]] != T(int(0)))
            goto next;
        continue;
       next:
        attend.emplace_back(make_pair(i, make_pair(j, k)));
      }
  sort(attend.begin(), attend.end());
  for(int i = 0; i < in0.size(); i ++) {
    in[i].resize(attend.size());
    for(int j = 0, cnt = 0; j < idx.size(); j ++)
      for(int k = 0; k < idx.size(); k ++)
        for(int m = 0; m < idx.size(); m ++)
          if(binary_search(attend.begin(), attend.end(),
              make_pair(j, make_pair(k, m))))
            in[i][cnt ++] =
              (in0[i][idx[j]][idx[k]][idx[m]] + T(int(1))) / T(int(2));
  }
  in0.resize(0);
  auto p(predv<T>(in, step));
  pair<SimpleSparseTensor<T>, SimpleSparseTensor<T> > res;
  for(int j = 0, cnt = 0; j < idx.size(); j ++)
    for(int k = 0; k < idx.size(); k ++)
      for(int m = 0; m < idx.size(); m ++)
        if(binary_search(attend.begin(), attend.end(),
             make_pair(j, make_pair(k, m)))) {
          res.first[idx[j]][idx[k]][idx[m]] =
            p.first[cnt] * T(int(2)) - T(int(1));
          res.second[idx[j]][idx[k]][idx[m]] =
            p.second[cnt ++] * T(int(2)) - T(int(1));
        }
  return res;
}

template <typename T> static inline vector<SimpleMatrix<T> > rgb2xyz(const vector<SimpleMatrix<T> >& rgb) {
  // CIE 1931 XYZ from wikipedia.org
  SimpleMatrix<T> mRGB2XYZ(3, 3);
  mRGB2XYZ(0, 0) = T(49000);
  mRGB2XYZ(0, 1) = T(31000);
  mRGB2XYZ(0, 2) = T(20000);
  mRGB2XYZ(1, 0) = T(17697);
  mRGB2XYZ(1, 1) = T(81240);
  mRGB2XYZ(1, 2) = T( 1063);
  mRGB2XYZ(2, 0) = T(0);
  mRGB2XYZ(2, 1) = T( 1000);
  mRGB2XYZ(2, 2) = T(99000);
  mRGB2XYZ /= T(17697);
  assert(rgb.size() == 3);
  assert(rgb[0].rows() == rgb[1].rows() && rgb[1].rows() == rgb[2].rows());
  assert(rgb[0].cols() == rgb[1].cols() && rgb[1].cols() == rgb[2].cols());
  auto xyz(rgb);
  xyz[0] = rgb[0] * mRGB2XYZ(0, 0) + rgb[1] * mRGB2XYZ(0, 1) + rgb[2] * mRGB2XYZ(0, 2);
  xyz[1] = rgb[0] * mRGB2XYZ(1, 0) + rgb[1] * mRGB2XYZ(1, 1) + rgb[2] * mRGB2XYZ(1, 2);
  xyz[2] = rgb[0] * mRGB2XYZ(2, 0) + rgb[1] * mRGB2XYZ(2, 1) + rgb[2] * mRGB2XYZ(2, 2);
  assert(xyz.size() == 3);
  assert(xyz[0].rows() == xyz[1].rows() && xyz[1].rows() == xyz[2].rows());
  assert(xyz[0].cols() == xyz[1].cols() && xyz[1].cols() == xyz[2].cols());
  return xyz;
}

template <typename T> static inline vector<SimpleMatrix<T> > xyz2rgb(const vector<SimpleMatrix<T> >& xyz) {
  // CIE 1931 XYZ from wikipedia.org
  SimpleMatrix<T> mRGB2XYZ(3, 3);
  mRGB2XYZ(0, 0) = T(49000);
  mRGB2XYZ(0, 1) = T(31000);
  mRGB2XYZ(0, 2) = T(20000);
  mRGB2XYZ(1, 0) = T(17697);
  mRGB2XYZ(1, 1) = T(81240);
  mRGB2XYZ(1, 2) = T( 1063);
  mRGB2XYZ(2, 0) = T(0);
  mRGB2XYZ(2, 1) = T( 1000);
  mRGB2XYZ(2, 2) = T(99000);
  mRGB2XYZ /= T(17697);
  const auto mXYZ2RGB(mRGB2XYZ.inverse());
  assert(xyz.size() == 3);
  assert(xyz[0].rows() == xyz[1].rows() && xyz[1].rows() == xyz[2].rows());
  assert(xyz[0].cols() == xyz[1].cols() && xyz[1].cols() == xyz[2].cols());
  auto rgb(xyz);
  rgb[0] = xyz[0] * mXYZ2RGB(0, 0) + xyz[1] * mXYZ2RGB(0, 1) + xyz[2] * mXYZ2RGB(0, 2);
  rgb[1] = xyz[0] * mXYZ2RGB(1, 0) + xyz[1] * mXYZ2RGB(1, 1) + xyz[2] * mXYZ2RGB(1, 2);
  rgb[2] = xyz[0] * mXYZ2RGB(2, 0) + xyz[1] * mXYZ2RGB(2, 1) + xyz[2] * mXYZ2RGB(2, 2);
  assert(rgb.size() == 3);
  assert(rgb[0].rows() == rgb[1].rows() && rgb[1].rows() == rgb[2].rows());
  assert(rgb[0].cols() == rgb[1].cols() && rgb[1].cols() == rgb[2].cols());
  return rgb;
}

static const vector<int>& pnTinySingle(const int& upper = 1) {
  static vector<int> pn;
  if(! pn.size()) pn.emplace_back(2);
  pn.reserve(upper);
  for(int i = pn.size(); i < upper; i ++) {
    for(int j = pn[pn.size() - 1] + 1; 0 <= j; j ++) {
      for(int k = 0; k < pn.size(); k ++)
        if(! (j % pn[k])) goto next_pn;
      pn.emplace_back(j);
      break;
     next_pn:
      ;
    }
  }
  return pn;
}

// N.B. start flip/flop
template <typename T> static inline SimpleMatrix<T> flip(const SimpleMatrix<T>& d) {
  auto res(d);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < d.rows(); i ++)
    res.row(res.rows() - 1 - i) = d.row(i);
  return res;
}

template <typename T> static inline SimpleMatrix<T> flop(const SimpleMatrix<T>& d) {
  auto res(d);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < d.cols(); i ++)
    res.setCol(res.cols() - 1 - i, d.col(i));
  return res;
}


// N.B. start isolate
template <typename T> static inline SimpleMatrix<T> harmlessSymmetrizeSquare(const SimpleMatrix<T>& m) {
  assert(0 < m.rows() && 0 < m.cols() && m.cols() == m.rows());
  SimpleMatrix<T> res(m.rows() + m.cols(), m.cols() + m.rows());
  res.I();
  return res.setMatrix(0, m.cols(), m.transpose() -
     SimpleMatrix<T>(m.rows(), m.rows()).I()).
    setMatrix(m.rows(), 0, flip<T>(flop<T>(m -
      SimpleMatrix<T>(m.cols(), m.cols()).I() )) );
}

template <typename T> static inline SimpleVector<T> balanceIntInvariant(const SimpleMatrix<T>& m, const vector<SimpleMatrix<T> >& db) {
  vector<SimpleMatrix<T> > logsymdb;
  logsymdb.reserve(db.size());
  for(int i = 0; i < db.size(); i ++) {
    logsymdb.emplace_back(log(harmlessSymmetrizeSquare(db[i])));
    assert(logsymdb[0].rows() == logsymdb[i].rows() &&
           logsymdb[0].cols() == logsymdb[i].cols());
  }
  auto mm(log(harmlessSymmetrizeSquare(m)));
  assert(logsymdb[0].rows() == mm.rows() &&
         logsymdb[0].cols() == mm.cols());
  SimpleMatrix<T> f(logsymdb.size(), logsymdb[0].rows() * logsymdb[0].cols());
  f.O();
  for(int i = 0; i < logsymdb.size(); i ++)
    for(int j = 0; j < logsymdb[i].rows(); j ++)
      for(int k = 0; k < logsymdb[i].cols(); k ++)
        f(i, j * logsymdb[i].cols() + k) = move(logsymdb[i](j, k));
  SimpleVector<T> vmm(mm.rows() * mm.cols());
  for(int j = 0; j < mm.rows(); j ++)
    for(int k = 0; k < mm.cols(); k ++)
      vmm[j * mm.cols() + k] = move(mm(j, k));
  return f.solveN(vmm);
}

template <typename T> static inline SimpleVector<T> powProgram(const pair<SimpleVector<T>, T>& m, const T& p, const vector<SimpleMatrix<T> >& db) {
  // N.B. power partial projected vector m by p using db.
  //      this is: tan ([a_0,...,a_n]^t x) == tan(S_0 ... x) form,
  //      getting (S_0...)^p.row(k) using program decomposition.
  // N.B. without db, we cannot get unique m^p in this invariant meaning.
  assert(db.size() && m.first.size() == db[0].cols() &&
         db[0].rows() == db[0].cols());
  for(int i = 1; i < db.size(); i ++)
    assert(db[i].rows() == db[0].rows() && db[i].cols() == db[0].cols());
  // N.B. balance inputs:
  // N.B. stub.
  assert(0 && "powProgram stub.");
  return m;
}

// N.B. start goki check
template <typename T> static inline bool less0(const T& x, const T& y) {
  return x.first[0] < y.first[0] || (x.first[0] == y.first[0] && x.first[1] < y.first[1]);
}

template <typename T> static inline bool lessf(const T& x, const T& y) {
  return x.first < y.first;
}

template <typename T> using triangles_t = pair<SimpleMatrix<T>, T>;

typedef enum {
  SHARPEN_X,
  SHARPEN_Y,
  SHARPEN_BOTH,
  ENLARGE_X,
  ENLARGE_Y,
  ENLARGE_BOTH,
  SHRINK_X,
  SHRINK_Y,
  SHRINK_BOTH,
  FLARGE_X,
  FLARGE_Y,
  FLARGE_BOTH,
  BLUR_X,
  BLUR_Y,
  BLUR_BOTH,
  COLLECT_BOTH,
  BUMP_BOTH,
  BLINK_X,
  BLINK_Y,
  BLINK_BOTH,
  REPRESENT,
  CLIP } direction_t;

template <typename T> bool saveobj(const vector<SimpleVector<T> >& data, const T& Mw0, const T& Mh0, const vector<SimpleVector<int> >& polys, const char* filename) {
  ofstream output;
  output.open(filename, std::ios::out);
  if(output.is_open()) {
    int lfs(0);
    const T Mh(Mh0 / T(2));
    const T Mw(Mw0 / T(2));
    for(int fslash(0) ; filename[fslash]; fslash ++)
      if(filename[fslash] == '/')
        lfs = fslash;
    if(lfs) lfs ++;
    output << "mtllib " << &filename[lfs] << ".mtl" << endl;
    output << "usemtl material0" << endl;
    for(int i = 0; i < data.size(); i ++)
      output << "v " << data[i][1] << " " << - data[i][0] << " " << data[i][2] << endl;
    for(int i = 0; i < data.size(); i ++)
      output << "vt " << data[i][1] / T(Mh) / T(2) << " " << T(1) - data[i][0] / T(Mw) / T(2) << endl;
    // xchg with clockwise/counter clockwise.
    for(int i = 0; i < polys.size(); i ++) {
      const int i0(polys[i][0] + 1);
      const int i1(polys[i][1] + 1);
      const int i2(polys[i][2] + 1);
      if(i0 != i1 && i1 != i2 && i2 != i0) {
        output << "f " << i0 << "/" << i0 << "/" << i0;
        output << " "  << i1 << "/" << i1 << "/" << i1;
        output << " "  << i2 << "/" << i2 << "/" << i2 << endl;
      }
    }
    output.close();
  } else {
    cerr << "Unable to open file for write: " << filename << endl;
    return false;
  }
  return true;
}

template <typename T> bool loadobj(vector<SimpleVector<T> >& data, vector<SimpleVector<int> >& polys, const char* filename) {
  ifstream input;
  input.open(filename);
  if(input.is_open()) {
    string work;
    while(getline(input, work) && !input.eof() && !input.bad()) {
      int i = 0;
      for( ; i < work.size() && work[i] == ' '; i ++);
      if(i + 1 < work.size() && work[i] == 'v' && work[i + 1] == ' ') {
        stringstream sub(work.substr(i + 2, work.size() - (i + 2)));
        SimpleVector<T> buf(3);
        sub >> buf[1];
        sub >> buf[0];
        sub >> buf[2];
        buf[0] = - buf[0];
        data.emplace_back(move(buf));
      } else if(i + 1 < work.size() && work[i] == 'f' && work[i + 1] == ' ') {
        stringstream sub(work.substr(i + 2, work.size() - (i + 2)));
        SimpleVector<int> wbuf(3);
        int  widx(0);
        bool flag(false);
        while(!sub.eof() && !sub.bad()) {
          sub >> wbuf[widx];
          if(wbuf[widx] >= 0)
            wbuf[widx] --;
          widx ++;
          if(widx > 2)
            flag = true;
          if(flag)
            polys.emplace_back(wbuf);
          widx %= 3;
          if(sub.eof() || sub.bad())
            break;
          sub.ignore(20, ' ');
        }
      }
    }
    for(int i = 0; i < polys.size(); i ++)
      for(int j = 0; j < polys[i].size(); j ++) {
        while(polys[i][j] < 0) polys[i][j] += data.size();
        polys[i][j] = polys[i][j] % data.size();
      }
    input.close();
  } else {
    cerr << "Unable to open file for read: " << filename << endl;
    return false;
  }
  return true;
}
  
template <typename T> bool saveMTL(const char* photo, const char* filename) {
  ofstream output;
  output.open(filename, std::ios::out);
  if(output.is_open()) {
    string pstr(photo);
    for(int i = 0; i < pstr.size(); i ++)
      if(pstr[pstr.size() - i - 1] == '.') {
        pstr = pstr.substr(0, max(int(0), int(pstr.size()) - i - 1)) + string(".ppm");
        break;
      }
    output << "newmtl material0" << endl;
    output << "Ka 1.000000 1.000000 1.000000" << endl;
    output << "Kd 1.000000 1.000000 1.000000" << endl;
    output << "Ks 0.000000 0.000000 0.000000" << endl;
    output << "illum 1" << endl;
    output << "map_Ka " << pstr << endl;
    output << "map_Kd " << pstr << endl << endl;
    output.close();
  } else {
    cerr << "Unable to open file for write: " << filename << endl;
    return false;
  }
  return true;
}

template <typename T> bool loaddat(const char* filename, string& header, vector<vector<T> >& data) {
  ifstream input;
  input.open(filename);
  if(input.is_open()) {
    string work;
    header = string("");
    data   = vector<vector<T> >();
    while(getline(input, work) && !input.eof() && !input.bad())
      if(whiteline(work))
        continue;
      else if(work[0] == ';')
        header += work + string("\n");
      else {
        stringstream ss(work);
        for(int i = 0, j = 0; ss.tellg() <= work.size(); j ++) {
          if(data.size() <= j)
            data.resize(j + 1, vector<T>());
          data[j].emplace_back(T(0));
          ss >> data[j][data[j].size() - 1];
        }
      }
    input.close();
  } else {
    cerr << "Unable to open file for read: " << filename << endl;
    return false;
  }
  return true;
}
  
template <typename T> bool savedat(const char* filename, string& header, vector<vector<T> >& data) {
  ofstream output;
  output.open(filename, std::ios::out);
  if(output.is_open()) {
    output << header;
    for(int i = 0; i < data[0].size(); i ++) {
      for(int j = 0; j < data.size(); j ++)
        output << (i < data[j].size() ? data[j][i] : T(0)) << " ";
      output << endl;
    }
    output.close();
  } else {
    cerr << "Unable to open file for write: " << filename << endl;
    return false;
  }
  return true;
}
  
template <typename T> bool loadcenterr(vector<SimpleVector<T> >& center, vector<T>& r, const char* filename) {
  center = vector<SimpleVector<T> >();
  r      = vector<T>();
  ifstream input;
  try {
    input.open(filename);
    string buf;
    while(getline(input, buf) && !input.eof() && !input.bad()) {
      stringstream sbuf(buf);
      SimpleVector<T> work(3);
      sbuf >> work[0];
      sbuf >> work[1];
      sbuf >> work[2];
      center.emplace_back(move(work));
      T workr;
      sbuf >> workr;
      r.emplace_back(workr);
    }
    input.close();
  } catch(...) {
    cerr << "Something had occured when reading center - r txt." << endl;
    return false;
  }
  return center.size() == r.size();
}

template <typename T> bool savecenterr(const char* filename, const vector<SimpleVector<T> >& center, const vector<T>& r) {
  ofstream output;
  output.open(filename, std::ios::out);
  if(output.is_open()) {
    assert(center.size() == r.size());
    for(int i = 0; i < center.size(); i ++) {
      assert(center[i].size() == 3);
      output << center[i][0] << " " << center[i][1] << " " << center[i][2] << " " << r[i] << endl;
    }
    output.close();
  } else {
    cerr << "Unable to open file for write: " << filename << endl;
    return false;
  }
  return true;
}

template <typename T> SimpleMatrix<T> sharpen(const int& size) {
  assert(0 < size);
  SimpleMatrix<T> s;
  const auto file(string("./.cache/lieonn/sharpen-") + to_string(size) +
#if defined(_FLOAT_BITS_)
    string("-") + to_string(_FLOAT_BITS_)
#else
    string("-ld")
#endif
  );
  ifstream cache(file.c_str());
  if(cache.is_open()) {
    cache >> s;
    cache.close();
  } else {
/*
    if(2 < size) {
      const auto s0(sharpen<T>(size - 1) * T(size - 1));
      s = SimpleMatrix<T>(size, size).O().setMatrix(0, 0, s0);
      s.setMatrix(1, 1, s.subMatrix(1, 1, size - 1, size - 1) + s0);
      s.row(0) *= T(2);
      s.row(s.rows() - 1) *= T(2);
      s /= T(2);
    } else
      s  = SimpleMatrix<T>(size, size).O();
*/
    auto dfts(dft<T>(size));
    static const auto Pi(atan(T(1)) * T(4));
    dfts.row(0) *= complex<T>(T(0));
    for(int i = 1; i < dfts.rows(); i ++) {
      // N.B. d/dt((d^(t)/dy^(t)) f), differential-integral space tilt on f.
      // DFTH.row(i) *= log(phase);
      // N.B. please refer enlarge.wxm, half freq space refer and uses each.
      //   -> This is sharpen operation at all because this is same as original
      //      picture when {x0 + x0.5, x0.5 + x1, x1 + x1.5, x1.5 + x2, ...}
      //      series, and both picture of dft is same, them, pick {x0, x1, ...}.
      dfts.row(i) /= exp(complex<T>(T(0), Pi * T(i) / T(dfts.rows()))) - complex<T>(T(1));
    }
    s = (dft<T>(- size) * dfts).template real<T>() / T(size - 1);
/*
    if(2 < size)
      s /= T(size);
*/
    ofstream ocache(file.c_str());
    ocache << s;
    ocache.close();
    cerr << "." << flush;
  }
  return s;
}

template <typename T> static inline SimpleMatrix<T> normalize(SimpleMatrix<T>& data, const T& upper = T(1)) {
  vector<SimpleMatrix<T> > work;
  work.emplace_back(move(data));
  auto res(normalize<T>(work, upper)[0]);
  data = move(work[0]);
  return res;
}

template <typename T> static inline SimpleMatrix<T> autoLevel(const SimpleMatrix<T>& data, const int& count = 0) {
  vector<SimpleMatrix<T> > work;
  work.emplace_back(data);
  return autoLevel(work, count)[0];
}

// N.B. this function is NOT thread safe.
template <typename T> SimpleMatrix<T> filter(const SimpleMatrix<T>& data, const direction_t& dir, const int& recur = 2, const int& rot = 0) {
  assert(0 <= rot);
  if(0 < rot) {
    auto res(filter<T>(data, dir, recur));
    if(rot <= 1) return res;
    for(int i = 0; i < rot; i ++) {
      cerr << "r" << flush;
      const auto theta((T(i) - T(rot - 1) / T(2)) * atan(T(1)) / (T(rot) / T(2)));
      res += center<T>(rotate<T>(filter<T>(rotate<T>(data, theta),
                       dir, recur), - theta), res);
    }
    return res /= T(rot + 1);
  }
  SimpleMatrix<T> result;
  static const auto Pi(atan2(T(1), T(1)) * T(4));
  switch(dir) {
  case SHARPEN_BOTH:
    result = filter<T>(filter<T>(data, SHARPEN_X, recur), SHARPEN_Y, recur);
    break;
  case ENLARGE_BOTH:
    result = filter<T>(filter<T>(data, ENLARGE_X, recur), ENLARGE_Y, recur);
    break;
  case SHRINK_BOTH:
    result = filter<T>(filter<T>(data, SHRINK_X, recur), SHRINK_Y, recur);
    break;
  case FLARGE_BOTH:
    result = filter<T>(filter<T>(data, FLARGE_X, recur), FLARGE_Y, recur);
    break;
  case BLUR_BOTH:
    result = filter<T>(filter<T>(data, BLUR_Y, recur), BLUR_X, recur);
    break;
  case BLINK_BOTH:
    result = filter<T>(filter<T>(data, BLINK_X, recur), BLINK_Y, recur);
    break;
  case SHARPEN_X:
    result = filter<T>(data.transpose(), SHARPEN_Y, recur).transpose();
    break;
  case ENLARGE_X:
    result = filter<T>(data.transpose(), ENLARGE_Y, recur).transpose();
    break;
  case SHRINK_X:
    result = filter<T>(data.transpose(), SHRINK_Y, recur).transpose();
    break;
  case FLARGE_X:
    result = filter<T>(data.transpose(), FLARGE_Y, recur).transpose();
    break;
  case BLUR_X:
    result = filter<T>(data.transpose(), BLUR_Y, recur).transpose();
    break;
  case BLINK_X:
    result = filter<T>(data.transpose(), BLINK_Y, recur).transpose();
    break;
  case BLUR_Y:
    {
      // N.B. d^exp(t)/dx^exp(t) f(x) == f(x + t dx), t != 0.
      //   so d^exp(- t)/dx^exp(- t)
      //   == d^(- exp(t))/dx^(- exp(t)) == f(x - t dx).
      //   we normalize with f(x - dx) + f(x + dx) in weak differential meaning.
      // Cor: d/dt d^t/d(x^t) f(x)
      //   == d/dt f(x + log(t) dx) == f'(x + log(t) dx) / g(x, t).
      // N.B. d^0/dx^0 f(x) == f(x)
      //   == d^exp(- inf)/dx^exp(- inf) f(x) == f(x - inf dx)
      //   == d^(- exp(- inf))/dx^(- exp(- inf dx))
      //   == d^(exp(inf))/dx^exp(inf) == f(x + inf dx)
      auto normalize(dft<T>(data.rows()) * data.template cast<complex<T> >());
      for(int i = 0; i < normalize.rows(); i ++) {
        const auto n(complex<T>(T(0), - T(2) * Pi * T(i) / T(normalize.rows())));
        normalize.row(i) *= exp(n) + exp(- n);
      }
      result = (dft<T>(- data.rows()) * normalize).template real<T>();
    }
    break;
  case SHARPEN_Y:
    result = filter(data - sharpen<T>(int(data.rows())) * data, CLIP);
    break;
  case ENLARGE_Y:
    {
      assert(2 <= data.rows());
      static vector<vector<SimpleMatrix<T> > > Eop;
      const auto& size(data.rows());
      if(Eop.size() <= size)
        Eop.resize(size + 1, vector<SimpleMatrix<T> >());
      else if(recur < Eop[size].size())
        goto eopi;
      if(Eop[size].size() <= recur)
        Eop[size].resize(recur + 1, SimpleMatrix<T>());
      {
        auto& eop(Eop[size][recur]);
        if(eop.cols() == size)
          goto eopi;
        cerr << "e" << flush;
        eop.resize((size - 1) * recur + 1, size);
        for(int j = 0; j < eop.rows(); j ++)
          // N.B. sampling th. hack isn't work well.
          eop.row(j) = taylor<T>(eop.cols(),
            (T(j) + T(int(1)) / T(int(2))) / T(eop.rows() - 1) * T(eop.cols() - 1),
            (T(j) - T(int(1)) / T(int(2))) / T(eop.rows() - 1) * T(eop.cols() - 1));
      }
     eopi:
      result = Eop[size][recur] * data;
    }
    break;
  case SHRINK_Y:
    {
      assert(2 <= data.rows());
      static vector<vector<SimpleMatrix<T> > > Eop;
      const auto& size(data.rows());
      if(Eop.size() <= size)
        Eop.resize(size + 1, vector<SimpleMatrix<T> >());
      else if(recur < Eop[size].size())
        goto sopi;
      if(Eop[size].size() <= recur)
        Eop[size].resize(recur + 1, SimpleMatrix<T>());
      {
        auto& eop(Eop[size][recur]);
        if(eop.cols() == size)
          goto sopi;
        cerr << "e" << flush;
        eop.resize(size / recur - 1, size);
        for(int j = 0; j < eop.rows(); j ++)
          eop.row(j) = taylor<T>(eop.cols(),
            (T(j) + T(int(1)) / T(int(2))) / T(eop.rows() - 1) * T(eop.cols() - 1),
            (T(j) - T(int(1)) / T(int(2))) / T(eop.rows() - 1) * T(eop.cols() - 1));
      }
     sopi:
      result = Eop[size][recur] * data;
    }
    break;
  case FLARGE_Y:
    {
      SimpleMatrix<T> work(data);
      for(int i = 0; i < work.rows(); i ++)
        for(int j = 0; j < work.cols(); j ++)
          work(i, j) += T(1) / T(256);
      Decompose<T> e(work.rows());
      result = SimpleMatrix<T>(work.rows() * recur, work.cols());
      for(int i = 0; i < work.cols(); i ++)
        result.setCol(i, e.enlarge(work.col(i), recur));
    }
    break;
  case COLLECT_BOTH:
    {
      const auto zy(diff<T>(data.rows()) * data);
      const auto zx(data * diff<T>(data.cols()).transpose());
      const auto zxx(diff<T>(data.rows()) * diff<T>(data.rows()) * data);
      const auto zxy(diff<T>(data.rows()) * data * diff<T>(data.cols()).transpose());
      const auto zyy(data * diff<T>(data.cols()).transpose() * diff<T>(data.cols()).transpose());
      result = SimpleMatrix<T>(data.rows(), data.cols()).O();
      for(int i = 0; i < result.rows(); i ++)
        for(int j = 0; j < result.cols(); j ++)
          // N.B. thanks to https://en.wikipedia.org/wiki/Gaussian_curvature .
          result(i, j) = abs(
            (zxx(i, j) * zyy(i, j) - zxy(i, j) * zxy(i, j)) /
            (zx(i, j) * zx(i, j) + zy(i, j) * zy(i, j) + T(int(1))) /
            (zx(i, j) * zx(i, j) + zy(i, j) * zy(i, j) + T(int(1))) );
    }
    break;
  case BUMP_BOTH:
    {
      SimpleMatrix<T> zscore(data.rows(), data.cols());
      result.resize(data.rows(), data.cols());
      result.O();
      zscore.O(- T(1));
      const auto rxy(T(min(data.rows(), data.cols())));
      const int  dratio(sqrt(sqrt(rxy)));
            SimpleVector<T> camera(2);
            SimpleVector<T> cpoint(2);
      camera[0] = T(0);
      camera[1] = T(1);
      cpoint[0] = T(1) / T(2 * dratio);
      for(int zi = 0; zi < dratio; zi ++) {
        cerr << "z" << flush;
        // N.B. projection scale is linear.
        cpoint[1] = T(zi) / T(dratio);
        // x-z plane projection of point p with camera geometry c to z=0.
        // c := camera, p := cpoint.
        // <c + (p - c) * t, [0, 1]> = 0
        const auto t(- camera[1] / (cpoint[1] - camera[1]));
        const auto y0((camera + (cpoint - camera) * t)[0] * rxy);
        if(abs(int(y0)) < 3 || rxy < abs(y0) * T(2)) continue;
        const auto Dop(diff<T>(abs(int(y0) & ~ int(1))));
        const auto Dop0((Dop.row(int(y0) / 2) + Dop.row(int(y0) / 2 + 1)) / T(2));
        const auto DDop(Dop * Dop);
        const auto DDop0((DDop.row(int(y0) / 2) + DDop.row(int(y0) / 2 + 1)) / T(2));
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
        for(int i = 0; i < data.rows(); i ++)
          for(int j = 0; j < data.cols(); j ++) {
            T fu(0), fv(0), L(0), M(0), N(0);
            for(int kk = 0; kk < Dop0.size(); kk ++) {
              fu += data(getImgPt<int>(i + kk - Dop0.size() / 2,
                data.rows()), j) * Dop0[kk];
              fv += data(i, getImgPt<int>(j + kk - Dop0.size() / 2,
                data.cols()) ) * Dop0[kk];
              L += data(getImgPt<int>(i + kk - Dop0.size() / 2,
                data.rows()), j) * DDop0[kk];
              N += data(i, getImgPt<int>(j + kk - Dop0.size() / 2,
                data.cols()) ) * DDop0[kk];
              for(int ll = 0; ll < Dop0.size(); ll ++)
                M += Dop0[kk] * Dop0[ll] *
                  data(getImgPt<int>(i + kk - Dop0.size() / 2, data.rows()),
                       getImgPt<int>(j + ll - Dop0.size() / 2, data.cols()));
            }
            // N.B. thanks to https://en.wikipedia.org/wiki/Gaussian_curvature .
            const auto lscore(abs((L * N - M * M) / (fu * fu + fv * fv + T(int(1))) / (fu * fu + fv * fv + T(int(1))) ));
            if(zscore(i, j) <= lscore) {
              result(i, j) = T(zi + 1) / T(dratio);
              zscore(i, j) = lscore;
            }
          }
      }
      // N.B. we don't need local to global with correct gaussian curvature.
      assert(result.rows() == data.rows() && result.cols() == data.cols());
    }
    break;
  case BLINK_Y:
    {
      auto dif(dft<T>(data.rows()) * data.template cast<complex<T> >());
      for(int i = 1; i < data.rows(); i ++)
        dif.row(i) *= - complex<T>(T(0), T(2)) * T(i) / T(data.rows());
      dif = dft<T>(- data.rows()) * dif;
      for(int i = 1; i < dif.rows() - 1; i ++)
        dif.row(i) += (dif.row(i - 1) + dif.row(i + 1)) * complex<T>(T(recur) / T(256));
      dif = dft<T>(data.rows()) * dif;
      for(int i = 1; i < data.rows(); i ++)
        dif.row(i) /= - complex<T>(T(0), T(2)) * T(i) / T(data.rows());
      result = (dft<T>(- data.rows()) * dif).template real<T>();
    }
    break;
  case REPRESENT:
    result = Decompose<T>(recur).represent(data, 2);
    break;
  case CLIP:
    result.resize(data.rows(), data.cols());
    for(int i = 0; i < result.rows(); i ++)
      for(int j = 0; j < result.cols(); j ++)
        result(i, j) = max(T(0), min(T(1), data(i, j)));
    break;
  default:
    assert(0 && "unknown command in filter (should not be reached.)");
  }
  return result;
}

template <typename T> class match_t {
public:
  typedef SimpleMatrix<T>   Mat;
  typedef SimpleVector<T>   Vec;
  typedef SimpleVector<int> Veci;
  Mat         rot;
  Vec         offset;
  T           ratio;
  T           rdepth;
  vector<int> dst;
  vector<int> src;
  T           thresh;
  T           othresh;
  inline match_t() {
    thresh  = T(0);
    othresh = T(0);
    initId();
  }
  inline match_t(const T& thresh, const T& d) {
    this->thresh  = thresh;
    this->othresh = thresh * d;
    initId();
  }
  inline match_t(const match_t<T>& s) {
    *this = s;
  }
  inline void initId() {
    rot       = Mat(3, 3);
    rot(0, 0) = rot(1, 1) = rot(2, 2) = T(1);
    rot(1, 0) = rot(2, 0) = rot(0, 1) = rot(2, 1)
              = rot(0, 2) = rot(1, 2) = T(0);
    offset    = Vec(3);
    offset[0] = offset[1] = offset[2] = T(0);
    ratio     = T(1);
    rdepth    = T(0);
  }
  inline match_t<T>  operator ~ () const {
    match_t<T> result(*this);
    result.rot    = rot.transpose();
    result.ratio  = T(1) / ratio;
    result.offset = - result.rot * offset * result.ratio;
    swap(result.src, result.dst);
    return result;
  }
  inline match_t<T>  operator / (const match_t<T>& s) const {
    match_t<T> result;
    result.rot    = rot   * s.rot.transpose();
    result.ratio  = ratio / s.ratio;
    result.offset = offset - result.rot * s.offset * result.ratio;
    result.rdepth = rdepth + s.rdepth;
    result.dst = vector<int>();
    result.src = vector<int>();
    for(int i = 0; i < src.size(); i ++)
      for(int j = 0; j < s.src.size(); j ++)
        if(src[i] == s.src[j]) {
          result.dst.emplace_back(  dst[i]);
          result.src.emplace_back(s.dst[j]);
        }
    return result;
  }
  inline match_t<T>& operator = (const match_t<T>& other) {
    rot        = other.rot;
    offset     = other.offset;
    ratio      = other.ratio;
    rdepth     = other.rdepth;
    dst  = other.dst;
    src  = other.src;
    thresh     = other.thresh;
    othresh    = other.othresh;
    return *this;
  }
  inline T distance(const match_t<T>& other, const Vec& p) {
    const auto d(transform(p) - other.transform(p));
    return sqrt(d.dot(d));
  }
  inline vector<Veci> hullConv(const vector<Veci>& srchull) const {
    assert(src.size() == dst.size());
    vector<Veci> res;
    res.reserve(srchull.size());
    for(int i = 0; i < srchull.size(); i ++) {
      Veci tmp(3);
      tmp[0] = tmp[1] = tmp[2] = - 1;
      for(int j = 0; j < srchull[i].size(); j ++)
        for(int k = 0; k < src.size(); k ++)
          if(src[k] == srchull[i][j]) {
            tmp[j] = dst[k];
            break;
          }
      assert(0 <= tmp[0] && 0 <= tmp[1] && 0 <= tmp[2]);
      res.emplace_back(tmp);
    }
    return res;
  }
  inline Vec transform(const Vec& x) const {
    return rot * x * ratio + offset;
  }
  inline vector<Vec> transform(const vector<Vec>& x) const {
    vector<Vec> result(x);
    for(int i = 0; i < result.size(); i ++)
      result[i] = transform(result[i]);
    return result;
  }
  inline bool operator < (const match_t<T>& x1) const {
    const T rratio(max(abs(   ratio), T(1) / abs(   ratio)));
    const T xratio(max(abs(x1.ratio), T(1) / abs(x1.ratio)));
    return dst.size() > x1.dst.size() || (dst.size() == x1.dst.size() && (rdepth < x1.rdepth || (rdepth == x1.rdepth && rratio < xratio)));
  }
  inline bool operator != (const match_t<T>& x) const {
    const auto test(offset - x.offset);
    const auto roterr(rot * x.rot.transpose());
    return !(abs(T(1) - roterr(0, 0)) <= thresh) ||
           !(abs(T(1) - roterr(1, 1)) <= thresh) ||
           !(abs(T(1) - roterr(2, 2)) <= thresh) ||
           !(sqrt(test.dot(test) / 
               (offset.dot(offset) + x.offset.dot(x.offset))) <= othresh) ||
           ratio * x.ratio < T(0) ||
           !(abs(ratio - x.ratio) / sqrt(ratio * x.ratio) <= thresh);
  }
  inline bool operator == (const match_t<T>& x) const {
    return ! (*this != x);
  }
  friend ostream& operator << (ostream& os, const match_t<T>& x) {
    os << x.rot;
    os << x.offset;
    os << x.ratio  << endl;
    os << x.rdepth << endl;
    assert(x.dst.size() == x.src.size());
    os << x.dst.size() << endl;
    for(int i = 0; i < x.dst.size(); i ++)
      os << x.dst[i] << " ";
    os << endl;
    for(int i = 0; i < x.src.size(); i ++)
      os << x.src[i] << " ";
    os << endl;
    os << x.thresh  << endl;
    return os;
  }
  friend istream& operator >> (istream& is, match_t<T>& x) {
    try {
      is >> x.rot;
      is >> x.offset;
      is >> x.ratio;
      is >> x.rdepth;
      int size(0);
      is >> size;
      assert(size > 0);
      x.dst.resize(size);
      x.src.resize(size);
      for(int i = 0; i < size; i ++)
        is >> x.dst[i];
      for(int i = 0; i < size; i ++)
        is >> x.src[i];
      is >> x.thresh;
    } catch(...) {
      assert(0 && "match_t input failed.");
    }
    return is;
  }
};

template <typename T> static inline pair<SimpleVector<T>, vector<SimpleVector<T> > > makeG(const vector<SimpleVector<T> >& in) {
  pair<SimpleVector<T>, vector<SimpleVector<T> > > res;
  res.second.reserve(in.size());
  assert(in.size() && in[0].size() == 3);
  res.first = in[0];
  for(int i = 1; i < in.size(); i ++)
    res.first += in[i];
  res.first /= T(in.size());
  for(int i = 0; i < in.size(); i ++)
    res.second.emplace_back(in[i] - res.first);
  assert(res.second.size() == in.size());
  return move(res);
}

template <typename T> static inline pair<T, vector<SimpleVector<T> > > normalizeG(const vector<SimpleVector<T> >& s) {
  pair<T, vector<SimpleVector<T> > > res;
  res.first = T(0);
  res.second.reserve(s.size());
  for(int i = 0; i < s.size(); i ++)
    res.first = max(max(res.first, abs(s[i][0])), max(abs(s[i][1]), abs(s[i][2])));
  for(int i = 0; i < s.size(); i ++)
    res.second.emplace_back(s[i] / res.first);
  assert(res.second.size() == s.size());
  return move(res);
}

template <typename T> static inline SimpleVector<T> toQuarterNormalize(const SimpleVector<T>& xyz) {
  assert(xyz.size() == 3);
  static const T    zero(int(0));
  static const auto twoPi(atan(T(int(1))) * T(int(4)));
  SimpleVector<T> quat(4);
  quat[0] = sqrt(xyz.dot(xyz) / T(int(6)));
  // y-z plane
  quat[1] = xyz[1] == zero && xyz[2] == zero ? zero : atan2(xyz[1], xyz[2]) / twoPi;
  // same for z-x.
  quat[2] = xyz[2] == zero && xyz[0] == zero ? zero : atan2(xyz[2], xyz[0]) / twoPi;
  // same for x-y.
  quat[3] = xyz[0] == zero && xyz[1] == zero ? zero : atan2(xyz[0], xyz[1]) / twoPi;
  return quat;
}

template <typename T> match_t<T> reconfigureMatch(match_t<T>& m, const vector<SimpleVector<T> >& dst0, const vector<SimpleVector<T> >& src0) {
  SimpleVector<T> off(3);
  off.O();
  for(int k = 0; k < m.dst.size(); k ++)
    off += dst0[m.dst[k]] - m.transform(src0[m.src[k]]);
  m.offset += (off /= T(int(m.dst.size())));
  if(m.dst.size() < 4) return m;
  SimpleMatrix<T> rot(3, 3);
  rot.I();
  for(int k = 0; k < m.dst.size() - 3; k ++) {
    SimpleMatrix<T> rotl(3, 3);
    SimpleMatrix<T> rotr(3, 3);
    rotl.O();
    rotr.O();
    for(int kk = 0; kk < 3; kk ++) {
      rotl.setCol(kk, dst0[m.dst[k + kk]]);
      rotr.setCol(kk, m.transform(src0[m.src[k + kk]]));
    }
    // dst == Q R == P R' == src, dst !~ avg(Q P^t) src'.
    rot *= rotl.QR().transpose() * rotr.QR();
  }
  m.rot  = pow(rot, T(1) / T(int(m.dst.size() - 3)));
  m.rot /= pow(abs(m.rot.determinant()), T(1) / T(3));
  off.O();
  for(int k = 0; k < m.dst.size(); k ++)
    off += dst0[m.dst[k]] - m.transform(src0[m.src[k]]);
  m.offset += (off /= T(int(m.dst.size())));
  T rlog0(int(0));
  for(int k = 0; k < m.dst.size(); k ++) {
    const auto& dstk(dst0[m.dst[k]]);
    const auto  srck(m.transform(src0[m.src[k]]));
    const auto  r(abs(dstk.dot(srck) / srck.dot(srck)));
    if(r != T(0)) rlog0 += log(r);
  }
  m.ratio *= exp(rlog0 / T(int(m.dst.size())));
  off.O();
  for(int k = 0; k < m.dst.size(); k ++)
    off += dst0[m.dst[k]] - m.transform(src0[m.src[k]]);
  m.offset += (off /= T(int(m.dst.size())));
  for(int k = 0; k < m.dst.size(); k ++) {
    const auto& dstk(dst0[m.dst[k]]);
    const auto  srck(m.transform(src0[m.src[k]]));
    const auto  err(dstk - srck);
    m.rdepth += sqrt(err.dot(err) / sqrt(dstk.dot(dstk) * srck.dot(srck)));
  }
  m.rdepth /= m.dst.size() * m.dst.size();
  return m;
}

template <typename T> vector<match_t<T> > matchPartialR(const vector<SimpleVector<T> >& dst0, const vector<SimpleVector<T> >& src0, const int& n = 1) {
  const auto  gs(normalizeG((makeG(dst0).second)));
  const auto  gp(normalizeG((makeG(src0).second)));
  const auto& dst(gs.second);
  const auto& src(gp.second);
  assert(dst.size() == dst0.size());
  assert(src.size() == src0.size());
  cerr << "match(" << dst.size() << ", " << src.size() << ")" << endl;
  SimpleMatrix<T> qdst(dst.size(), 4);
  SimpleMatrix<T> qsrc(src.size(), 4);
  for(int i = 0; i < qdst.rows(); i ++)
    qdst.row(i) = toQuarterNormalize<T>(dst[i]);
  for(int i = 0; i < qsrc.rows(); i ++)
    qsrc.row(i) = toQuarterNormalize<T>(src[i]);
  vector<SimpleVector<T> > test;
  vector<pair<int, int> > idx;
  test.reserve(qdst.rows() * qsrc.rows());
  idx.reserve(qdst.rows() * qsrc.rows());
  for(int i = 0; i < qdst.rows(); i ++)
    for(int j = 0; j < qsrc.rows(); j ++) {
      idx.emplace_back(make_pair(i, j));
      test.emplace_back(qdst.row(i) - qsrc.row(j));
    }
  const auto cr(crush<T>(test, test[0].size(), n));
  vector<match_t<T> > mm;
  mm.reserve(cr.size());
  for(int i = 0; i < cr.size(); i ++) {
    if(! cr[i].first.size()) continue;
    match_t<T> m(T(int(1)) / T(int(100)), max(gs.first, gp.first));
    SimpleVector<int> dfix, sfix;
    dfix.resize(dst.size());
    sfix.resize(src.size());
    dfix.I(false);
    sfix.I(false);
    m.dst.reserve(min(dst.size(), src.size()));
    m.src.reserve(min(dst.size(), src.size()));
    auto avg(cr[i].first[0]);
    for(int j = 1; j < cr[i].first.size(); j ++)
      avg += cr[i].first[j];
    avg /= T(int(cr[i].first.size()));
    vector<pair<T, int> > pp;
    pp.reserve(cr[i].first.size());
    for(int j = 0; j < cr[i].first.size(); j ++) {
      const auto err(cr[i].first[j] - avg);
      pp.emplace_back(make_pair(err.dot(err), j));
    }
    sort(pp.begin(), pp.end());
    for(int j = 0; j < pp.size(); j ++) {
      const auto& lidx(idx[cr[i].second[pp[j].second]]);
      if(dfix[lidx.first] || sfix[lidx.second]) continue;
      dfix[lidx.first] = sfix[lidx.second] = true;
      m.dst.emplace_back(lidx.first);
      m.src.emplace_back(lidx.second);
    }
    if(m.dst.size() < 4) continue;
    m.dst.reserve(m.dst.size());
    m.src.reserve(m.src.size());
    m = reconfigureMatch<T>(m, dst0, src0);
    for(int i = 0; i < m.rot.rows(); i ++)
      for(int j = 0; j < m.rot.cols(); j ++)
        if(! isfinite(m.rot(i, j))) goto nofix;
    for(int i = 0; i < m.offset.size(); i ++)
      if(! isfinite(m.offset[i])) goto nofix;
    if(! isfinite(m.ratio)) goto nofix;
    mm.emplace_back(move(m));
    cerr << mm[mm.size() - 1] << endl;
   nofix:
    ;
  }
  sort(mm.begin(), mm.end());
  return mm;
}

template <typename T> static inline vector<match_t<T> > matchPartial(const vector<SimpleVector<T> >& dst0, const vector<SimpleVector<T> >& src0, const int& n = 1) {
  auto m(matchPartialR<T>(src0, dst0, n));
  for(int i = 0; i < m.size(); i ++)
    m[i] = ~ m[i];
  return m;
}


template <typename T> void drawMatchLine(SimpleMatrix<T>& map, const SimpleVector<T>& lref0, const SimpleVector<T>& lref1, const T& c) {
  int idxm(0);
  int idxM(1);
  if(abs(lref1[idxM] - lref0[idxM]) < abs(lref1[idxm] - lref0[idxm]))
    swap(idxm, idxM);
  const auto d10(lref1 - lref0);
  const auto dlt(abs(lref0[idxM] - lref1[idxM]));
  if(dlt == T(0)) return;
  const auto denom(T(1) / dlt);
  for(int i = 0; i <= int(ceil(dlt)); i ++) {
    const auto gidx(lref0 + d10 * T(i) * denom);
    map(max(int(0), min(int(gidx[0]), int(map.rows() - 1))),
        max(int(0), min(int(gidx[1]), int(map.cols() - 1)))) = c;
  }
  return;
}

template <typename T> void drawMatchTriangle(SimpleMatrix<T>& map, SimpleVector<T> lref0, SimpleVector<T> lref1, SimpleVector<T> lref2, const T& c) {
  // make middle point to lref2 on index 0.
  if((lref0[0] <= lref1[0] && lref1[0] <= lref2[0]) ||
     (lref2[0] <= lref1[0] && lref1[0] <= lref0[0]))
    swap(lref1, lref2);
  if((lref1[0] <= lref0[0] && lref0[0] <= lref2[0]) ||
     (lref2[0] <= lref0[0] && lref0[0] <= lref1[0]) )
    swap(lref0, lref2);
  const auto d0(lref0 - lref2);
  const auto d1(lref1 - lref2);
  const auto d2(lref1 - lref0);
  const auto idx(abs(d2[0]) < abs(d2[1]) ? 1 : 0);
  if(abs(d0[idx]) != T(int(0)) && abs(d2[idx]) != T(int(0)))
    try {
      for(int i = 0; i <= int(abs(d0[idx])); i ++)
        drawMatchLine<T>(map, d0 * (abs(d0[idx]) - T(i)) / abs(d0[idx]) + lref2,
                              d2 * T(i) / abs(d2[idx]) + lref0, c);
    } catch (const char* e) {
      ; /* fall through */
    }
  if(abs(d1[idx]) != T(int(0)) && abs(d2[idx]) != T(int(0)))
    try {
      for(int i = 0; i <= int(abs(d1[idx])); i ++)
        drawMatchLine<T>(map, d1 * (abs(d1[idx]) - T(i)) / abs(d1[idx]) + lref2,
                            - d2 * T(i) / abs(d2[idx]) + lref1, c);
    } catch (const char* e) {
      ; /* fall through */
    }
  return;
}

template <typename T> void addMeshTri(vector<SimpleVector<int> >& res, vector<pair<SimpleVector<T>, int> >& scan, const vector<SimpleVector<T> >& p, const int& idx) {
  assert(0 <= idx && idx < scan.size());
  vector<int> elim;
  if(0 <= idx - 1 &&
     scan[idx].first[0] < scan[idx - 1].first[0] &&
     scan[idx].first[0] < scan[idx + 1].first[0]) {
    elim.emplace_back(idx);
    SimpleVector<int> lres(3);
    lres[0] = scan[idx - 1].second;
    lres[1] = scan[idx].second;
    lres[2] = scan[idx + 1].second;
    bool psize(false);
    for(int k = 0; k < 3; k ++)
      psize = psize || p.size() <= lres[k];
    if(! psize)
      res.emplace_back(move(lres));
  }
  if(idx + 3 < scan.size() &&
     scan[idx + 2].first[0] < scan[idx + 1].first[0] &&
     scan[idx + 2].first[0] < scan[idx + 3].first[0]) {
    elim.emplace_back(idx + 2);
    SimpleVector<int> lres(3);
    lres[0] = scan[idx + 1].second;
    lres[1] = scan[idx + 2].second;
    lres[2] = scan[idx + 3].second;
    bool psize(false);
    for(int k = 0; k < 3; k ++)
      psize = psize || p.size() <= lres[k];
    if(! psize)
      res.emplace_back(move(lres));
  }
  {
    SimpleVector<int> lres(3);
    lres[0] = scan[idx].second;
    lres[1] = scan[idx + 1].second;
    lres[2] = scan[idx + 2].second;
    bool psize(false);
    for(int k = 0; k < 3; k ++)
      psize = psize || p.size() <= lres[k];
    if(! psize)
      res.emplace_back(move(lres));
  }
  sort(elim.begin(), elim.end());
  for(int j = 0; j < elim.size(); j ++)
    scan.erase(scan.begin() + elim[j] - j);
  return;
}

template <typename T> vector<SimpleVector<int> > mesh2(const vector<SimpleVector<T> >& p, const vector<int>& pp) {
  vector<pair<SimpleVector<T>, int> > sp;
  sp.reserve(pp.size());
  T Mxy(int(0));
  for(int i = 0; i < p.size(); i ++)
    Mxy = max(Mxy, max(abs(p[i][0]), abs(p[i][1])));
  Mxy *= T(int(2));
  SimpleMatrix<T> lrot(3, 3);
  lrot.I();
  lrot(0, 0) =    lrot(1, 1) = cos(T(int(1)) / max(Mxy, T(pp.size())));
  lrot(0, 1) = - (lrot(1, 0) = sin(T(int(1)) / max(Mxy, T(pp.size()))));
  T    m1((lrot * p[pp[0]])[1]);
  auto M1(m1);
  for(int i = 0; i < pp.size(); i ++) {
    sp.emplace_back(make_pair(lrot * p[pp[i]], pp[i]));
    sp[i].first[2] = T(0);
    m1 = min(m1, sp[i].first[1]);
    M1 = max(M1, sp[i].first[1]);
  }
  sort(sp.begin(), sp.end(), less0<pair<SimpleVector<T>, int> >);
  vector<pair<SimpleVector<T>, int> > scan;
  scan.reserve(sp.size() + 2);
  scan.emplace_back(sp[0]);
  scan[scan.size() - 1].first[0] -= T(2);
  scan[scan.size() - 1].first[1]  = m1 - T(1);
  scan.emplace_back(sp[0]);
  scan[scan.size() - 1].first[0] -= T(1);
  scan[scan.size() - 1].first[1]  = M1 + T(1);
  vector<SimpleVector<int> > res;
  res.reserve(sp.size());
  for(int i = 0; i < sp.size(); i ++) {
    // N.B. lrot support this on lattice.
    assert(! i || (sp[i].first[0] != sp[i - 1].first[0] &&
                   sp[i].first[1] != sp[i - 1].first[1]) );
    // scanline update
    int idx;
    for(idx = 0; idx < scan.size(); idx ++)
      if(sp[i].first[1] < scan[idx].first[1]) break;
    idx = max(int(0), min(int(scan.size()) - 2, idx - 1));
    assert(scan[idx].first[1] < sp[i].first[1]);
    assert(sp[i].first[1] < scan[idx + 1].first[1]);
    assert(scan[idx].first[0] < sp[i].first[0]);
    assert(scan[idx + 1].first[0] < sp[i].first[0]);
    scan.insert(scan.begin() + idx + 1, pair<SimpleVector<T>, int>(sp[i]));
    assert(scan[idx].first[1] < scan[idx + 1].first[1] &&
           scan[idx + 1].first[1] < scan[idx + 2].first[1]);
    addMeshTri<T>(res, scan, p, idx);
  }
  while(6 < scan.size()) {
    const auto before(scan.size());
    for(int i = 2; i < scan.size() - 3; i ++)
      addMeshTri<T>(res, scan, p, i);
    if(before == scan.size()) break;
  }
  res.reserve(res.size());
  for(int i = 0; i < res.size(); i ++)
    if(p[res[i][0]][0] * p[res[i][1]][1]
     + p[res[i][1]][0] * p[res[i][2]][1]
     + p[res[i][2]][0] * p[res[i][0]][1]
     - p[res[i][0]][1] * p[res[i][1]][0]
     - p[res[i][1]][1] * p[res[i][2]][0]
     - p[res[i][2]][1] * p[res[i][0]][0] < T(0))
      swap(res[i][0], res[i][1]);
  return res;
}

template <typename T> static inline vector<SimpleVector<int> > mesh2(const vector<SimpleVector<T> >& p) {
  vector<int> pp;
  pp.reserve(p.size());
  for(int i = 0; i < p.size(); i ++) pp.emplace_back(i);
  return mesh2<T>(p, pp);
}

// get bump with multiple scale and vectorized result.
template <typename T> vector<SimpleVector<T> > getTileVec(const SimpleMatrix<T>& in, const int& vbox = 1) {
  vector<SimpleVector<T> > geoms;
  geoms.reserve((in.rows() / vbox + 1) * (in.cols() / vbox + 1));
  // N.B. align with BUMP_BOTH z-axis rxy.
  const auto diag(sqrt(sqrt(T(min(in.rows(), in.cols())) )) );
  for(int i = 0; i < in.rows() / vbox + 1; i ++)
    for(int j = 0; j < in.cols() / vbox + 1; j ++) {
      if(in.rows() < (i + 1) * vbox ||
         in.cols() < (j + 1) * vbox) {
        SimpleVector<T> gbuf(3);
        gbuf[0] = T(i * vbox);
        gbuf[1] = T(j * vbox);
        gbuf[2] = geoms[geoms.size() - 1][2];
        geoms.emplace_back(gbuf);
      } else {
        SimpleVector<T> work(3);
        work[0] = T(i * vbox);
        work[1] = T(j * vbox);
        work[2] = diag * in(i * vbox, j * vbox);
        geoms.emplace_back(work);
      }
    }
  T avg(int(0));
  for(int i = 0; i < geoms.size(); i ++)
    avg += geoms[i][2];
  avg /= T(geoms.size());
  for(int i = 0; i < geoms.size(); i ++)
    geoms[i][2] -= avg;
  geoms.reserve(geoms.size());
  return geoms;
}

template <typename T> vector<SimpleVector<T> > getHesseVec(const SimpleMatrix<T>& in, const int& vbox = 300) {
  const auto guard(max(int(1), int(sqrt(T(in.rows() * in.cols() / vbox)))));
  vector<SimpleVector<T> > geoms;
  geoms.reserve(vbox + 4);
  const auto x(in * diff<T>(in.cols()).transpose());
  const auto y(diff<T>(in.rows()) * in);
  const auto xx(in * diff<T>(in.cols()).transpose() * diff<T>(in.cols()).transpose());
  const auto xy(diff<T>(in.rows()) * in * diff<T>(in.cols()).transpose());
  const auto yy(diff<T>(in.rows()) * diff<T>(in.rows()) * in);
  // N.B. align with BUMP_BOTH z-axis rxy.
  const auto diag(sqrt(sqrt(T(min(in.rows(), in.cols())) )) );
  vector<pair<T, pair<int, int> > > score;
  score.reserve(in.rows() * in.cols());
  for(int i = 0; i < in.rows(); i ++)
    for(int j = 0; j < in.cols(); j ++)
      score.emplace_back(make_pair(abs((xx(i, j) * yy(i, j) - xy(i, j) * xy(i, j)) / ((T(int(1)) + x(i, j)) * (T(int(1)) + y(i, j)) - x(i, j) * y(i, j))), make_pair(i, j)));
  sort(score.begin(), score.end());
  vector<pair<int, int> > cache;
  cache.reserve(score.size());
  for(int i = score.size() - 1; 0 <= i && geoms.size() < abs(vbox); i --)
    if(! binary_search(cache.begin(), cache.end(),
           make_pair(score[i].second.first / guard,
                     score[i].second.second / guard)) ) {
      SimpleVector<T> g(3);
      g[0] = T(int(score[i].second.first));
      g[1] = T(int(score[i].second.second));
      g[2] = diag *
        in(score[i].second.first, score[i].second.second);
      geoms.emplace_back(move(g));
      cache.emplace_back(make_pair(score[i].second.first / guard,
                                   score[i].second.second / guard));
      sort(cache.begin(), cache.end());
    }
  SimpleVector<T> g(3);
  g[0] = T(int(0));
  g[1] = T(int(0));
  g[2] = diag * in(0, 0);
  geoms.emplace_back(g);
  g[0] = T(int(in.rows() - 1));
  g[1] = T(int(0));
  g[2] = diag * in(in.rows() - 1, 0);
  geoms.emplace_back(g);
  g[0] = T(int(0));
  g[1] = T(int(in.cols() - 1));
  g[2] = diag * in(0, in.cols() - 1);
  geoms.emplace_back(g);
  g[0] = T(int(in.rows() - 1));
  g[1] = T(int(in.cols() - 1));
  g[2] = diag * in(in.rows() - 1, in.cols() - 1);
  geoms.emplace_back(g);
  T avg(int(0));
  for(int i = 0; i < geoms.size(); i ++)
    avg += geoms[i][2];
  avg /= T(geoms.size());
  for(int i = 0; i < geoms.size(); i ++)
    geoms[i][2] -= avg;
  geoms.reserve(geoms.size());
  return geoms;
}

template <typename T> SimpleMatrix<T> tilt(const SimpleMatrix<T>& in, vector<triangles_t<T> >& triangles, const T& depth = - T(10000)) {
  cerr << "t" << flush;
  SimpleMatrix<T> result(in.rows(), in.cols());
  result.O();
  vector<pair<T, triangles_t<T>> > zbuf;
  zbuf.resize(triangles.size(), make_pair(T(0), triangles_t<T>()));
  assert(zbuf.size() == triangles.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int j = 0; j < triangles.size(); j ++) {
    auto& tri(triangles[j]);
    assert(tri.first.rows() == tri.first.cols() && tri.first.rows() == 3);
    // N.B. /= 3 isn't needed because only the order is the matter.
    // N.B.  = - ... isn't matches the .obj output.
    //      also this matches bump output.
    zbuf[j].first  = tri.first(0, 2) + tri.first(1, 2) + tri.first(2, 2);
    zbuf[j].second = move(tri);
  }
  sort(zbuf.begin(), zbuf.end(), lessf<pair<T, triangles_t<T> > >);
  int i;
  // XXX: patent???
  // N.B. we could avoid with this because no z-buffer matrix on them,
  //      but this is obscure.
  for(i = 0; i < zbuf.size() && zbuf[i].first < depth; i ++) ;
  for( ; i < zbuf.size(); i ++) {
    const auto& zbi(zbuf[i].second);
    drawMatchTriangle<T>(result, zbi.first.row(0), zbi.first.row(1), zbi.first.row(2), zbi.second);
  }
  return result;
}

template <typename T> static inline SimpleMatrix<T> tilt(const SimpleMatrix<T>& in, const vector<triangles_t<T> >& triangles, const T& depth = - T(10000)) {
  auto tris(triangles);
  return tilt<T>(in, tris, depth);
}

template <typename T> vector<triangles_t<T> > triangles(const SimpleMatrix<T>& in, const SimpleMatrix<T>& bump, const match_t<T>& m) {
  assert(in.rows() == bump.rows() && in.cols() == bump.cols());
  auto points(getTileVec<T>(bump));
  auto facets(mesh2<T>(points));
  vector<triangles_t<T> > triangles;
  triangles.resize(facets.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < facets.size(); i ++) {
    triangles_t<T> work;
    work.first = SimpleMatrix<T>(3, 3);
    for(int j = 0; j < 3; j ++) {
      assert(0 <= facets[i][j] && facets[i][j] < points.size());
      work.first.row(j) = m.transform(points[facets[i][j]]);
    }
    for(int j = 0; j < 2; j ++) {
      if(work.first(0, j) <= work.first(1, j) && work.first(0, j) <= work.first(2, j))
        work.first(0, j) = floor(work.first(0, j));
      else if(work.first(1, j) <= work.first(0, j) && work.first(1, j) <= work.first(2, j))
        work.first(1, j) = floor(work.first(1, j));
      else if(work.first(2, j) <= work.first(0, j) && work.first(2, j) <= work.first(1, j))
        work.first(2, j) = floor(work.first(2, j));
      if(work.first(1, j) <= work.first(0, j) && work.first(2, j) <= work.first(0, j))
        work.first(0, j) = ceil(work.first(0, j));
      else if(work.first(0, j) <= work.first(1, j) && work.first(2, j) <= work.first(1, j))
        work.first(1, j) = ceil(work.first(1, j));
      else if(work.first(0, j) <= work.first(2, j) && work.first(1, j) <= work.first(2, j))
        work.first(2, j) = ceil(work.first(2, j));
    }
    if(T(0) <= points[facets[i][0]][0] && points[facets[i][0]][0] < T(in.rows()) &&
       T(0) <= points[facets[i][0]][1] && points[facets[i][0]][1] < T(in.cols()))
      work.second = in(int(points[facets[i][0]][0]),
                       int(points[facets[i][0]][1]));
    else
      work.second = T(0);
    triangles[i] = move(work);
  }
  return triangles;
}

template <typename T> SimpleMatrix<T> draw(const SimpleMatrix<T>& img, const vector<SimpleVector<T> >& shape, const vector<SimpleVector<T> >& emph, const vector<SimpleVector<int> >& hull) {
  assert(shape.size() == emph.size());
  vector<triangles_t<T> > tris;
  tris.resize(hull.size());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < hull.size(); i ++) {
    assert(hull[i].size() == 3);
    assert(0 <= hull[i][0] && hull[i][0] < shape.size());
    assert(0 <= hull[i][1] && hull[i][1] < shape.size());
    assert(0 <= hull[i][2] && hull[i][2] < shape.size());
    triangles_t<T> work;
    work.first = SimpleMatrix<T>(3, 3);
    for(int j = 0; j < 3; j ++)
      work.first.row(j) = emph[hull[i][j]];
    work.second = img(max(int(0), min(int(img.rows() - 1),
                          int(shape[hull[i][0]][0]))),
                      max(int(0), min(int(img.cols() - 1),
                          int(shape[hull[i][0]][1]))));
    tris[i] = move(work);
  }
  return tilt<T>(img * T(0), tris);
}

template <typename T> SimpleMatrix<T> draw(const SimpleMatrix<T>& img, const vector<SimpleVector<T> >& shape, const vector<SimpleVector<int> >& hull, const bool& elim = false) {
  auto result(img);
  T    M(0);
  T    m(0);
  for(int i = 0; i < shape.size(); i ++) {
    if(i) {
      M = max(M, shape[i][2]);
      m = min(m, shape[i][2]);
    } else
      M = m = shape[i][2];
  }
  auto tsrc(shape);
  if(M - m != T(0))
    for(int i = 0; i < tsrc.size(); i ++)
      tsrc[i][2] = elim ? T(0) : (tsrc[i][2] - m) / (M - m);
  for(int ii = 0; ii < hull.size(); ii ++)
    drawMatchTriangle<T>(result, tsrc[hull[ii][0]],
                                 tsrc[hull[ii][1]],
                                 tsrc[hull[ii][2]],
                                 (tsrc[hull[ii][0]][2] + tsrc[hull[ii][1]][2] +
                                  tsrc[hull[ii][2]][2]) / T(3));
  return result;
}

template <typename T> static inline vector<SimpleVector<T> > takeShape(const vector<SimpleVector<T> >& dst, const vector<SimpleVector<T> >& src, const match_t<T>& match, const T& ratio) {
  auto result(dst);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < match.src.size(); i ++)
    result[match.dst[i]] += (match.transform(src[match.src[i]]) - dst[match.dst[i]]) * ratio;
  return result;
}

template <typename T> static inline SimpleMatrix<T> showMatch(const SimpleMatrix<T>& dstimg, const vector<SimpleVector<T> >& dst, const vector<SimpleVector<int> >& hull, const T& emph = T(1)) {
  auto map(dstimg);
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int k = 0; k < hull.size(); k ++) {
    drawMatchLine<T>(map, dst[hull[k][0]], dst[hull[k][1]], emph);
    drawMatchLine<T>(map, dst[hull[k][1]], dst[hull[k][2]], emph);
    drawMatchLine<T>(map, dst[hull[k][2]], dst[hull[k][0]], emph);
  }
  return map;
}

template <typename T> static inline SimpleMatrix<T> makeRefMatrix(const SimpleMatrix<T>& orig, const int& start) {
  SimpleMatrix<T> result(orig.rows(), orig.cols());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < orig.rows() * orig.cols(); i ++)
    result(i % orig.rows(), i / orig.rows()) = i + start;
  return result;
}

template <typename T> static inline SimpleMatrix<T> pullRefMatrix(const SimpleMatrix<T>& ref, const int& start, const SimpleMatrix<T>& orig) {
  assert(orig.rows() == ref.rows() && orig.cols() == ref.cols());
  SimpleMatrix<T> result(ref.rows(), ref.cols());
  result.O();
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int i = 0; i < ref.rows() * ref.cols(); i ++) {
    const int ly(i % ref.rows());
    const int lx(i / ref.rows());
    const int v(int(ref(ly, lx)) - start);
    if(0 <= v && v < orig.rows() * orig.cols())
      result(ly, lx) = orig(v % orig.rows(), v / orig.rows());
    else
      result(ly, lx) = T(0);
  }
  return result;
}

template <typename T> SimpleMatrix<T> reShape(const SimpleMatrix<T>& cbase, const SimpleMatrix<T>& vbase, const int& count, const T& thresh) {
  assert(cbase.rows() && cbase.cols() && vbase.rows() && vbase.cols());
  assert(cbase.rows() == vbase.rows() && cbase.cols() == vbase.cols());
  vector<pair<T, pair<int, int> > > vpoints;
  vpoints.reserve(vbase.rows() * vbase.cols());
  for(int i = 0; i < vbase.rows(); i ++)
    for(int j = 0; j < vbase.cols(); j ++)
      vpoints.emplace_back(make_pair(vbase(i, j), make_pair(i, j)));
  sort(vpoints.begin(), vpoints.end());
  SimpleMatrix<T> res(vbase.rows(), vbase.cols());
  T avg(0);
  for(int i = 0, ii = 0; i < vpoints.size(); i ++) {
    if(abs(vpoints[i].first - vpoints[ii].first) < abs(vpoints[vpoints.size() - 1].first - vpoints[0].first) / T(count) && i < vpoints.size() - 1)
      avg += cbase(vpoints[i].second.first, vpoints[i].second.second);
    else if(i != ii) {
      if(i == vpoints.size() - 1) {
        avg += cbase(vpoints[i].second.first, vpoints[i].second.second);
        i ++;
      }
      avg /= i - ii;
      for(int j = ii; j < i; j ++)
        res(vpoints[j].second.first, vpoints[j].second.second) = avg;
      avg  = T(0);
      ii   = i;
    }
  }
  SimpleMatrix<T> masker(res);
  SimpleMatrix<T> mask(res.rows(), res.cols());
  mask.O(false);
  for(int i = 0; i < mask.rows(); i ++)
    for(int j = 0; j < mask.cols(); j ++)
      if(! mask(i, j)) {
        vector<pair<int, int> > store;
        vector<pair<int, int> > tries;
        tries.emplace_back(make_pair(+ 1,   0));
        tries.emplace_back(make_pair(  0, + 1));
        tries.emplace_back(make_pair(- 1,   0));
        tries.emplace_back(make_pair(  0, - 1));
        vector<pair<int, int> > stack;
        stack.emplace_back(make_pair(i, j));
        T   avg(0);
        int cnt(0);
        while(stack.size()) {
          const auto pop(stack[stack.size() - 1]);
          stack.pop_back();
          const int& yy(pop.first);
          const int& xx(pop.second);
          if(0 <= yy && yy < mask.rows() && 0 <= xx && xx < mask.cols() &&
             masker(i, j) == masker(yy, xx) &&
             abs(cbase(i, j) - cbase(yy, xx)) <= thresh &&
             ! mask(yy, xx)) {
            mask(yy, xx) = true;
            store.emplace_back(make_pair(yy, xx));
            for(int ii = 0; ii < tries.size(); ii ++)
              stack.emplace_back(make_pair(yy + tries[ii].first, xx + tries[ii].second));
            avg += cbase(yy, xx);
            cnt ++;
          }
        }
        avg /= T(cnt);
        for(int i = 0; i < store.size(); i ++)
          res(store[i].first, store[i].second) = avg;
      }
  return res;
}

template <typename T> SimpleMatrix<T> reColor(const SimpleMatrix<T>& cbase, const SimpleMatrix<T>& vbase, const int& count, const T& intensity) {
  assert(cbase.rows() && cbase.cols() && vbase.rows() && vbase.cols());
  vector<pair<T, pair<int, int> > > vpoints;
  vector<pair<T, pair<int, int> > > cpoints;
  vpoints.reserve(vbase.rows() * vbase.cols());
  cpoints.reserve(cbase.rows() * cbase.cols());
  for(int i = 0; i < vbase.rows(); i ++)
    for(int j = 0; j < vbase.cols(); j ++)
      vpoints.emplace_back(make_pair(vbase(i, j), make_pair(i, j)));
  for(int i = 0; i < cbase.rows(); i ++)
    for(int j = 0; j < cbase.cols(); j ++)
      cpoints.emplace_back(make_pair(cbase(i, j), make_pair(i, j)));
  sort(vpoints.begin(), vpoints.end());
  sort(cpoints.begin(), cpoints.end());
  SimpleVector<T> vv(vpoints.size());
  SimpleVector<T> cc(cpoints.size());
  for(int i = 0; i < vv.size(); i ++)
    vv[i] = vpoints[i].first;
  for(int i = 0; i < cc.size(); i ++)
    cc[i] = cpoints[i].first;
  const auto ccc(Decompose<T>(count).mimic(cc, vv, intensity));
        auto res(cbase);
  for(int i = 0; i < ccc.size(); i ++)
    res(cpoints[i].second.first, cpoints[i].second.second) = ccc[i];
  return res;
}

template <typename T> SimpleMatrix<T> reColor3(const SimpleMatrix<T>& ccbase, const SimpleMatrix<T>& vbase, const int& count) {
  assert(ccbase.rows() && ccbase.cols() && vbase.rows() && vbase.cols());
  auto cbase(ccbase);
  for(int i = 0; i < cbase.rows(); i ++)
    for(int j = 0; j < cbase.cols(); j ++)
      cbase(i, j) += T(int(1)) / T(int(256));
  vector<pair<T, pair<int, int> > > vpoints;
  vector<pair<T, pair<int, int> > > cpoints;
  vpoints.reserve(vbase.rows() * vbase.cols());
  cpoints.reserve(cbase.rows() * cbase.cols());
  for(int i = 0; i < vbase.rows(); i ++)
    for(int j = 0; j < vbase.cols(); j ++)
      vpoints.emplace_back(make_pair(vbase(i, j), make_pair(i, j)));
  for(int i = 0; i < cbase.rows(); i ++)
    for(int j = 0; j < cbase.cols(); j ++)
      cpoints.emplace_back(make_pair(cbase(i, j), make_pair(i, j)));
  sort(vpoints.begin(), vpoints.end());
  sort(cpoints.begin(), cpoints.end());
  SimpleMatrix<T> res(cbase.rows(), cbase.cols());
  for(int i = 0; i < count; i ++) {
    T scorev(0);
    T scorec(0);
    const auto vstart(i * int(vpoints.size() / count));
    const auto vend(min((i + 1) * int(vpoints.size() / count), int(vpoints.size())));
    const auto cstart(i * int(cpoints.size() / count));
    const auto cend(min((i + 1) * int(cpoints.size() / count), int(cpoints.size())));
    for(int j = vstart; j < vend; j ++)
      scorev += vpoints[j].first;
    for(int j = cstart; j < cend; j ++)
      scorec += cpoints[j].first;
    scorev /= T(vend - vstart);
    scorec /= T(cend - cstart);
    for(int j = cstart; j < cend; j ++)
      res(cpoints[j].second.first, cpoints[j].second.second) =
        cpoints[j].first * scorev / scorec;
  }
  return res;
}

template <typename T> SimpleMatrix<T> reColor(const SimpleMatrix<T>& cbase, const int& count, const T& intensity) {
  assert(cbase.rows() && cbase.cols());
  vector<pair<T, pair<int, int> > > cpoints;
  cpoints.reserve(cbase.rows() * cbase.cols());
  for(int i = 0; i < cbase.rows(); i ++)
    for(int j = 0; j < cbase.cols(); j ++)
      cpoints.emplace_back(make_pair(cbase(i, j), make_pair(i, j)));
  sort(cpoints.begin(), cpoints.end());
  SimpleVector<T> cc(cpoints.size());
  for(int i = 0; i < cc.size(); i ++)
    cc[i] = cpoints[i].first;
  const auto ccc(Decompose<T>(count).emphasis(cc, intensity));
        auto res(cbase);
  for(int i = 0; i < ccc.size(); i ++)
    res(cpoints[i].second.first, cpoints[i].second.second) = ccc[i];
  return res;
}

template <typename T> vector<vector<int> > catImage(const vector<SimpleMatrix<T> >& imgs, const int& cs = 40) {
  for(int i = 1; i < imgs.size(); i ++) {
    assert(imgs[i].rows() == imgs[0].rows());
    assert(imgs[i].cols() == imgs[0].cols());
  }
  vector<SimpleVector<T> > work;
  work.reserve(imgs.size());
  for(int i = 0; i < imgs.size(); i ++) {
    work.emplace_back(SimpleVector<T>(imgs[i].rows() * imgs[i].cols()).O());
    for(int j = 0; j < imgs[i].rows(); j ++)
      work[i].setVector(imgs[i].cols() * j, imgs[i].row(j));
  }
  auto cg(crush<T>(work, cs, 0));
  vector<vector<int> > res;
  res.reserve(cg.size());
  for(int i = 0; i < cg.size(); i ++)
    res.emplace_back(move(cg[i].second));
  return res;
}

template <typename T> static inline SimpleMatrix<T> rgb2d(const vector<SimpleMatrix<T> > rgb) {
  auto xyz(rgb2xyz<T>(rgb));
  SimpleMatrix<T> result(rgb[0].rows(), rgb[0].cols());
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
  for(int j = 0; j < rgb[0].rows(); j ++) {
    for(int k = 0; k < rgb[0].cols(); k ++)
      result(j, k) = sqrt(xyz[0](j, k) * xyz[0](j, k) + xyz[1](j, k) * xyz[1](j, k) + xyz[2](j, k) * xyz[2](j, k));
  }
  return result;
}

template <typename T> static inline match_t<T> tiltprep(const SimpleMatrix<T>& in, const int& idx, const int& samples, const T& psi, const T& z0 = T(int(0))) {
  const auto Pi(atan2(T(int(1)), T(int(1))) * T(int(4)));
  const auto theta(T(2) * Pi * T(idx) / T(samples));
  const auto lpsi(Pi * psi);
  SimpleMatrix<T> R0(3, 3);
  SimpleMatrix<T> R1(3, 3);
  R0(0, 0) =   cos(theta);
  R0(0, 1) =   sin(theta);
  R0(0, 2) = 0.;
  R0(1, 0) = - sin(theta);
  R0(1, 1) =   cos(theta);
  R0(1, 2) = 0.;
  R0(2, 0) = 0.;
  R0(2, 1) = 0.;
  R0(2, 2) = 1.;
  R1(0, 0) = 1.;
  R1(0, 1) = 0.;
  R1(0, 2) = 0.;
  R1(1, 0) = 0.;
  R1(1, 1) = 1.;
  R1(1, 2) = 0.;
  R1(2, 0) = 0.;
  R1(2, 1) = 0.;
  R1(2, 2) = 1.;
  R1(0, 0) =   cos(lpsi);
  R1(0, 2) = - sin(lpsi);
  R1(2, 0) =   sin(lpsi);
  R1(2, 2) =   cos(lpsi);
  match_t<T> m;
  m.rot    = R0.transpose() * R1 * R0;
  SimpleVector<T> pcenter(3);
  pcenter[0] = T(in.rows() - 1) / T(2);
  pcenter[1] = T(in.cols() - 1) / T(2);
  pcenter[2] = z0;
  // x -> m.rot * x, same center
  // x - origin -> m.rot * (x - origin)
  // x -> m.rot * x - m.rot * origin + origin.
  m.offset = pcenter - m.rot * pcenter;
  m.ratio  = T(1);
  return m;
}


// N.B. start corpus without corpus class which frequently updated.
template <typename T> static inline bool equalStrClip(const T& a, const T& b) {
  int cmp(0), jidx(0);
  for( ; !cmp && jidx < min(a.size(), b.size()); jidx ++)
    cmp = a[jidx] ^ b[jidx];
  return !cmp && min(a.size(), b.size()) <= jidx;
}

template <typename T> static inline bool lessEqualStrClip(const T& a, const T& b) {
  return a < b ||  equalStrClip<T>(a, b);
}

template <typename T> static inline bool lessNotEqualStrClip(const T& a, const T& b) {
  return a < b && !equalStrClip<T>(a, b);
}

template <typename T> class gram_t {
public:
  T           str;
  vector<int> rptr;
  inline gram_t() {
    this->str  = T();
    this->rptr = vector<int>();
  }
  inline ~gram_t() { ; }
  inline gram_t(const gram_t<T>& x) { *this = x; }
  inline gram_t& operator = (const gram_t<T>& x) {
    str  = x.str;
    rptr = x.rptr;
    return *this;
  }
  inline bool operator == (const gram_t<T>& x) const {
    return ! (*this != x);
  }
  inline bool operator != (const gram_t<T>& x) const {
    return str != x.str || rptr != x.rptr;
  }
  inline bool operator < (const gram_t<T>& x1) const {
    return str < x1.str;
  }
};

template <typename T> static inline bool lessCount(const gram_t<T>& dst, const gram_t<T>& src) {
  return (dst.rptr.size() < src.rptr.size()) ||
    (dst.rptr.size() == src.rptr.size() && dst.str < src.str);
}

template <typename T, typename U> class lword {
public:
  inline lword(const int& loop) {
    this->dicts.resize(loop, vector<gram_t<U> >());
  }
  inline ~lword() { ; }
  
  vector<gram_t<U> > compute(const U& input);

private:
  inline bool       isin(const U& key) {
    assert(key.size() < dicts.size());
    const vector<gram_t<U> >& dict(dicts[key.size()]);
    gram_t<U> key0;
    key0.str = key;
    auto p(lower_bound(dict.begin(), dict.end(), key0));
    return dict.begin() <= p && p < dict.end() && p->str == key;
  }

  inline gram_t<U>& find(const U& key) {
    static gram_t<U> dummy;
    assert(key.size() < dicts.size());
    vector<gram_t<U> >& dict(dicts[key.size()]);
    gram_t<U> key0;
    key0.str = key;
    auto p(lower_bound(dict.begin(), dict.end(), key0));
    if(p < dict.begin() || dict.end() <= p || p->str != key) {
      assert(0 && "slipping find.");
      return dummy;
    }
    return *p;
  }

  inline void       assign(const gram_t<U>& val) {
    assert(val.str.size() < dicts.size());
    vector<gram_t<U> >& dict(dicts[val.str.size()]);
    auto p(lower_bound(dict.begin(), dict.end(), val));
    if(val.rptr.size()) {
      // delete duplicates:
      gram_t<U> work;
      work.str = val.str;
      auto& vptr(work.rptr = val.rptr);
      sort(vptr.begin(), vptr.end());
      vptr.erase(unique(vptr.begin(), vptr.end()), vptr.end());
      if(p < dict.begin() || dict.end() <= p || p->str != work.str) {
        dict.emplace_back(work);
        sort(dict.begin(), dict.end());
      } else
        *p = work;
    } else if(dict.begin() <= p && p < dict.end() && p->str == val.str)
      dict.erase(p);
    return;
  }

  vector<T>                   dict0;
  vector<vector<gram_t<U> > > dicts;
};

template <typename T, typename U> vector<gram_t<U> > lword<T, U>::compute(const U& input) {
  cerr << "lword(" << dicts.size() << ", " << input.size() << ")" << endl;
  // bigram
  map<U, vector<int> > mapw;
  for(int i = 1; i < input.size(); i ++) {
    U work;
    work += T(input[i - 1]);
    work += T(input[i]);
    mapw[work].emplace_back(i);
    if(!binary_search(dict0.begin(), dict0.end(), input[i])) {
      dict0.emplace_back(input[i]);
      sort(dict0.begin(), dict0.end());
    }
  }
  for(auto itr = mapw.begin(); itr != mapw.end(); ++ itr) {
    gram_t<U> work;
    work.str = itr->first;
    work.rptr.insert(work.rptr.end(), itr->second.begin(), itr->second.end());
    assign(work);
  }
  // construct
  int i;
  for(i = 2; i < dicts.size() - 1; i ++) {
    map<U, vector<int> > amap;
#if defined(_OPENMP)
#pragma omp parallel for schedule(static, 1)
#endif
    for(auto itr = dicts[i].begin(); itr < dicts[i].end(); ++ itr) {
      const gram_t<U>& idxkey(*itr);
      U key2;
      for(int j = 1; j < idxkey.str.size(); j ++)
        key2 += idxkey.str[j];
      key2 += T(' ');
      for(auto itr2 = dict0.begin(); itr2 != dict0.end(); ++ itr2) {
        key2[key2.size() - 1] = T(*itr2);
        if(!isin(key2))
          continue;
        const gram_t<U>& idxkey2(find(key2));
        U workkey(idxkey.str);
        workkey += *itr2;
        vector<int> idxwork;
        int tt = 0;
        int ss = 0;
        while(tt < idxkey2.rptr.size() && ss < idxkey.rptr.size()) {
          if(idxkey2.rptr[tt] == idxkey.rptr[ss] + 1) {
            idxwork.emplace_back(idxkey.rptr[ss]);
            ss ++;
            tt ++;
          } else if(idxkey2.rptr[tt] >= idxkey.rptr[ss])
            ss ++;
          else
            tt ++;
        }
        assert(idxwork.size() <= min(idxkey.rptr.size(), idxkey2.rptr.size()));
        if(idxwork.size() < 2) continue;
#if defined(_OPENMP)
#pragma omp critical
#endif
        {
          amap[workkey].insert(amap[workkey].end(), idxwork.begin(), idxwork.end());
        }
      }
    }
    // construct next stage.
    for(auto itr = amap.begin(); itr != amap.end(); ++ itr) {
      gram_t<U> work;
      work.str  = itr->first;
      work.rptr.insert(work.rptr.end(), itr->second.begin(), itr->second.end());
      assign(work);
    }
  }
  
  // count up.
  vector<gram_t<U> > words;
  for(int i = 0; i < dicts.size(); i ++)
    words.insert(words.end(), dicts[i].begin(), dicts[i].end());
  dict0 = vector<T>();
  dicts = vector<vector<gram_t<U> > >();
  sort(words.begin(), words.end());
  vector<gram_t<U> > result;
  result.reserve(words.size());
  for(int i = 0; i < words.size() - 1; i ++) {
    const auto mw(min(words[i].str.size(), words[i + 1].str.size()));
    if(! (words[i].str.substr(0, mw) == words[i + 1].str.substr(0, mw) &&
          words[i].rptr == words[i + 1].rptr) ) {
      if(words[i].str.size() < words[i + 1].str.size() &&
         words[i].str.substr(1, mw) == words[i + 1].str.substr(0, mw) &&
         words[i].rptr.size() == words[i + 1].rptr.size()) {
        int j;
        for(j = 0; j < words[i].rptr.size(); j ++)
          if(words[i].rptr[j] != words[i + 1].rptr[j] + 1)
            break;
        if(j != words[i].rptr.size())
          result.emplace_back(move(words[i]));
      } else if(words[i].str.size() > words[i + 1].str.size() &&
         words[i].str.substr(0, mw) == words[i + 1].str.substr(1, mw) &&
         words[i].rptr.size() == words[i + 1].rptr.size()) {
        int j;
        for(j = 0; j < words[i].rptr.size(); j ++)
          if(words[i].rptr[j] + 1 != words[i + 1].rptr[j])
            break;
        if(j != words[i].rptr.size())
          result.emplace_back(move(words[i]));
      } else
        result.emplace_back(move(words[i]));
   }
  }
  result.emplace_back(move(words[words.size() - 1]));
  return result;
}

template <typename T, typename U> static inline U getCut(const U& input, const int& idx, const int& szwindow) {
  if(idx < 0 || input.size() <= idx * szwindow / 2)
    return U("XXX: getCut");
  return input.substr(idx * szwindow / 2, szwindow);
}

template <typename T, typename U> static inline ostream& outTagged(ostream& os, const U& name, const corpus<T, U>& cstat, const int& idx, const T& score, const U& input, const int& szwindow) {
  os << "<span id=\"" << name << idx << "\">";
  os << "score: " << score << " : ";
  os << getCut<T, U>(input, idx, szwindow);
  os << "<input class=\"gatherdetail\" type=\"checkbox\"><div class=\"gatherdetail\">";
  os << cstat.reverseLink().second << "<br/>";
  os << cstat.getAttributed(words);
  os << "</div></span>";
  return os;
}

template <typename T, typename U> static inline void getAbbreved(corpus<T, U>& cstat, const vector<U>& detailtitle, const vector<U>& detail, const vector<U>& delimiter) {
  assert(detailtitle.size() == detail.size());
  for(int i = 0; i < detail.size(); i ++)
    cstat = cstat.abbrev(detailtitle[i], corpus<T, U>(detail[i], delimiter));
  return;
}

template <typename T, typename U> static inline bool getDetailed(corpus<T, U>& cstat, const U& input, const int& idx, const vector<U>& detailtitle, const vector<U>& detail, const vector<U>& delimiter, const int& szwindow, const T& thresh) {
  assert(detailtitle.size() == detail.size());
  if(idx < 0 || input.size() <= idx * szwindow / 2)
    return false;
  assert(0 <= idx && idx * szwindow / 2 < input.size());
  cstat = corpus<T, U>(getCut<T>(input, idx, szwindow), delimiter).simpleThresh(thresh);
  for(int i = 0; i < detail.size(); i ++)
    cstat = cstat.withDetail(detailtitle[i], corpus<T, U>(detail[i], delimiter), thresh).simpleThresh(thresh);
  return true;
}

template <typename T, typename U> ostream& preparedTOC(ostream& os, const U& input, const vector<U>& detailtitle, const vector<U>& detail, const vector<U>& topictitle, const vector<U>& topics, const vector<U>& delimiter, const int& szwindow, const int& outblock, const int& nrwords, const T& redig = T(1), const bool& reverse = false) {
  assert(detailtitle.size() == detail.size());
  assert(topictitle.size()  == topics.size());
  os << "prepTOC: " << flush;
  if(!topics.size())
    return os << "zero input. <br />";
  vector<corpus<T, U> > istats;
  T threshin(int(0));
  for(int i = - int(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) );
          i <= 0; i ++) {
    threshin = T(int(1)) - pow(T(int(2)), - T(abs(i)));
    vector<int> idx;
    istats.resize(0);
    istats.resize(input.size() / (szwindow / 2));
    for(int j = 0; j < istats.size(); j ++) {
      getDetailed<T, U>(istats[j], input, j, detailtitle, detail, delimiter, szwindow, threshin);
      istats[j].reDig(redig);
      istats[j].absfy();
      auto lidx(istats[j].countIdx());
      idx.insert(idx.end(), lidx.begin(), lidx.end());
    }
    sort(idx.begin(), idx.end());
    idx.erase(unique(idx.begin(), idx.end()), idx.end());
    cerr << threshin << " : " << idx.size() << endl;
    if(nrwords <= idx.size()) break;
  }
  for(int i = 0; i < topics.size(); i ++) {
    vector<pair<T, pair<int, int> > > topicidx;
    vector<corpus<T, U> > stats;
    stats.resize(input.size() / (szwindow / 2));
    for(int j = 0; j < stats.size(); j ++) {
      getDetailed<T, U>(stats[j], topics[i], j, detailtitle, detail, delimiter, szwindow, threshin);
      stats[j].reDig(redig);
      stats[j].absfy();
    }
    for(int j = 0; j < istats.size(); j ++) {
      int idx(0);
      T   score(0);
      for(int k = 0; k < stats.size(); k ++) {
        const auto  lscore(reverse ? T(1) / abs(istats[j].prej(stats[k]))
                                   :            istats[j].prej(stats[k]) );
        if(isfinite(lscore) && score <= lscore) {
          idx   = k;
          score = lscore;
        }
      }
      topicidx.emplace_back(make_pair(- score, make_pair(j, idx)));
    }
    sort(topicidx.begin(), topicidx.end());
    for(int j = 0; j < topicidx.size(); j ++) {
      const auto& stat0(istats[topicidx[j].second.first]);
      const auto& stat1(stats[topicidx[j].second.second]);
      if(outblock < j)
        break;
      os << topictitle[i] << " ";
      outTagged<T,U>(os, U("prepTOC0_") + to_string(i) + U("_"), stat0, topicidx[j].second.first, topicidx[j].first, input, szwindow);
      os << "<br/>";
      outTagged<T,U>(os, U("prepTOC_")  + to_string(i) + U("_"), stat1, topicidx[j].second.second, topicidx[j].first, topics[i], szwindow);
      os << "<br/><br/>" << endl;
    }
    os << "<br/><br/>" << endl;
  }
  return os;
}

template <typename T, typename U> ostream& optimizeTOC(ostream& os, const U& input, const vector<U>& detail, const vector<U>& detailtitle, const vector<U>& delimiter, const int& szwindow, const int& outblock, const int& nrwords, const T& redig = T(1), const bool& countnum = false, const U& notcheck = U("")) {
  assert(notcheck == U(""));
  os << "optTOC: " << flush;
  SimpleSparseMatrix<T> scores;
  int Midx(0);
  vector<corpus<T, U> > stats;
  T   threshin(int(0));
  for(int i = - int(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) );
          i <= 0; i ++) {
    threshin = T(int(1)) - pow(T(int(2)), - T(abs(i)));
    vector<int> idx;
    stats.resize(0);
    stats.resize(input.size() / (szwindow / 2));
    for(int j = 0; j < stats.size(); j ++) {
      getDetailed<T, U>(stats[j], input, j, detailtitle, detail, delimiter, szwindow, threshin);
      stats[j].reDig(redig);
      stats[j].absfy();
      auto lidx(stats[j].countIdx());
      idx.insert(idx.end(), lidx.begin(), lidx.end());
    }
    sort(idx.begin(), idx.end());
    idx.erase(unique(idx.begin(), idx.end()), idx.end());
    cerr << threshin << " : " << idx.size() << endl;
    if(nrwords <= idx.size()) break;
  }
  for(int i = 0; i < stats.size(); i ++)
    for(int j = i + 1; j < stats.size(); j ++) {
      scores[i][j] = - stats[i].prej(stats[j]);
      Midx = max(Midx, j);
    }
  os << "." << flush;
  int idx(0);
  vector<int> phrases;
  for(int ii = 0; phrases.size() < Midx; ii ++) {
    vector<vector<pair<T, pair<int, int> > > > lscore;
    int  lidx(0);
    T    Mscore(0);
    bool fixed(false);
    for(int i = 0; i < Midx; i ++)
      if(!binary_search(phrases.begin(), phrases.end(), i)) {
        vector<pair<T, pair<int, int> > > llscore;
        for(int j = 0; j < Midx; j ++)
          if(!binary_search(phrases.begin(), phrases.end(), j) &&
             llscore.size() <= outblock)
            llscore.emplace_back(make_pair(scores[min(i, j)][max(i, j)], make_pair(i, j)));
        sort(llscore.begin(), llscore.end());
        lscore.emplace_back(llscore);
        bool ok(false);
        if(countnum) {
          T lllscore(0);
          for(int j = 0; j < llscore.size(); j ++)
            lllscore += T(1);
          ok = Mscore < lllscore;
          Mscore = max(Mscore, lllscore);
        } else {
          T lllscore(0);
          for(int j = 0; j < llscore.size(); j ++)
            lllscore -= llscore[j].first;
          ok = Mscore < lllscore;
          if(fixed)
            Mscore = max(Mscore, lllscore);
          else {
            Mscore = lllscore;
            fixed  = true;
          }
        }
        if(ok)
          lidx  = i;
      } else
        lscore.emplace_back(vector<pair<T, pair<int, int> > >());
    if(lscore[lidx].size() <= 0) {
      for(int i = 0; i < lscore.size(); i ++)
        if(lscore[i].size()) {
          lidx = i;
          break;
        }
      if(! lscore[lidx].size())
        break;
    }
    phrases.emplace_back(lscore[lidx][0].second.first);
    sort(phrases.begin(), phrases.end());
    const auto& cs(stats[lscore[lidx][0].second.first]);
    for(int i = 0; i < lscore[lidx].size(); i ++)
      phrases.emplace_back(lscore[lidx][i].second.second);
    sort(phrases.begin(), phrases.end());
    os << "<form action=\"../../../../puts.php\" method=\"POST\"><div>";
    os << "base : ";
    outTagged<T,U>(os, U("optTOC0_"), cs, lscore[lidx][0].second.first, lscore[lidx][0].first, input, szwindow) << "<br/>";
    os << "<br/>Show/Hide : <input class=\"gather\" type=\"checkbox\"><div class=\"gather\">";
    for(int i = 0; i < lscore[lidx].size(); i ++)
      outTagged<T,U>(os, U("optTOC_"), stats[lscore[lidx][i].second.second], lscore[lidx][i].second.second, lscore[lidx][i].first, input, szwindow) << "<br/>";
    os << "</div></div><textarea name=\"entry\">";
    os << getCut<T,U>(input, lscore[lidx][0].second.first, szwindow);
    os << "\n";
    for(int i = 0; i < lscore[lidx].size(); i ++)
      os << getCut<T,U>(input, lscore[lidx][i].second.second, szwindow) << "\n";
    os << "</textarea>";
    os << "<input type=\"hidden\" name=\"name\" value=\"append\" />";
    os << "<input type=\"hidden\" name=\"adddict\" value=\"\" />";
    os << "<input type=\"submit\" value=\"Append\" />";
    os << "</form><br/>" << endl;
  }
  return os << endl;
}

template <typename T, typename U> ostream& diff(ostream& os, const U& input, const vector<U>& detail0, const vector<U>& detailtitle0, const vector<U>& detail1, const vector<U>& detailtitle1, const vector<U>& delimiter, const int& szwindow, const int& outblock, const int& nrwords, const T& redig = T(1), const bool& same = false) {
  assert(detail0.size() == detailtitle0.size() &&
         detail1.size() == detailtitle1.size());
  os << "diff:" << flush;
  T threshin(int(0));
  for(int i = - int(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) );
          i <= 0; i ++) {
    threshin = T(int(1)) - pow(T(int(2)), - T(abs(i)));
    corpus<T, U> stat;
    vector<int> idx;
    for(int j = 0; j < input.size() / (szwindow / 2); j ++) {
      getDetailed<T, U>(stat, input, j, detailtitle0, detail0, delimiter, szwindow, threshin);
      stat.reDig(redig);
      stat.absfy();
      auto lidx(stat.countIdx());
      idx.insert(idx.end(), lidx.begin(), lidx.end());
    }
    for(int j = 0; j < input.size() / (szwindow / 2); j ++) {
      getDetailed<T, U>(stat, input, j, detailtitle1, detail1, delimiter, szwindow, threshin);
      stat.reDig(redig);
      stat.absfy();
      auto lidx(stat.countIdx());
      idx.insert(idx.end(), lidx.begin(), lidx.end());
    }
    sort(idx.begin(), idx.end());
    idx.erase(unique(idx.begin(), idx.end()), idx.end());
    cerr << threshin << " : " << idx.size() << endl;
    if(nrwords <= idx.size()) break;
  }
  corpus<T, U> cstat, dstat;
  vector<pair<T, int> > scores;
  for(int i = 0; ; i ++) {
    if(!getDetailed<T, U>(cstat, input, i, detailtitle0, detail0, delimiter, szwindow, threshin) ||
       !getDetailed<T, U>(dstat, input, i, detailtitle1, detail1, delimiter, szwindow, threshin))
      break;
    getAbbreved<T, U>(cstat, detailtitle1, detail1, delimiter);
    getAbbreved<T, U>(dstat, detailtitle0, detail0, delimiter);
    cstat.reDig(redig);
    dstat.reDig(redig);
    const auto score(cstat.prej(dstat));
    os << score << ":" << flush;
    if(isfinite(score))
      scores.emplace_back(make_pair(same ? - score : score, i));
  }
  sort(scores.begin(), scores.end());
  for(int ii = 0; ii < scores.size() && ii <= outblock; ii ++) {
    const T&   score(scores[ii].first);
    const int& i(scores[ii].second);
    if(!getDetailed<T, U>(cstat, input, i, detailtitle0, detail0, delimiter, szwindow, threshin) ||
       !getDetailed<T, U>(dstat, input, i, detailtitle1, detail1, delimiter, szwindow, threshin))
      break;
    getAbbreved<T, U>(cstat, detailtitle1, detail1, delimiter);
    getAbbreved<T, U>(dstat, detailtitle0, detail0, delimiter);
    cstat.reDig(redig);
    dstat.reDig(redig);
    auto diff(cstat - dstat);
    diff.reDig(redig);
    os << "score: " << score << " : " << diff.serialize() << "<br/>" << endl;
    outTagged<T,U>(os, U("optTOC_src"), cstat, i, score, input, szwindow) << "<br/>" << endl;
    outTagged<T,U>(os, U("optTOC_dst"), dstat, i, score, input, szwindow) << "<br/>" << endl;
    outTagged<T,U>(os, U("optTOC_diff"), diff, i, score, input, szwindow) << "<br/><br/>" << endl;
  }
  return os << endl;
}

template <typename T> static inline vector<T> cutText(const T& input, const vector<T>& eliminate, const vector<T>& delimiter, const bool& f_sort = false) {
  vector<T> result;
  T         workbuf;
  for(int i = 0; i < input.size(); i ++) {
    workbuf += input[i];
    for(int j = 0; j < delimiter.size(); j ++)
      if(workbuf.size() >= delimiter[j].size() &&
         workbuf.substr(workbuf.size() - delimiter[j].size(), delimiter[j].size()) == delimiter[j]) {
        if(workbuf.size() - delimiter[j].size())
          result.emplace_back(workbuf.substr(0, workbuf.size() - delimiter[j].size()));
        workbuf = T();
        goto next;
      }
    for(int j = 0; j < eliminate.size(); j ++)
      if(workbuf.size() >= eliminate[j].size() &&
        workbuf.substr(workbuf.size() - eliminate[j].size(), eliminate[j].size()) == eliminate[j]) {
        workbuf = workbuf.substr(0, workbuf.size() - eliminate[j].size());
        break;
      }
   next:
    ;
  }
  if(workbuf.size())
    result.emplace_back(workbuf);
  if(f_sort)
    sort(result.begin(), result.end());
  return result;
}

template <typename T, typename U> static inline SimpleVector<T> countWords(const U& orig, const vector<U>& words) {
  SimpleVector<T> result(words.size());
  for(int i = 0; i < result.size(); i ++)
    result[i] = T(0);
  for(int i = 0; i < orig.size(); i ++) {
    const U work(orig.substr(i, orig.size() - i - 1));
    for(int j = 0; j < words.size(); j ++)
      if(equalStrClip<U>(work, words[j]) && work.size() >= words[j].size())
        result[j] = T(1);
  }
  return result;
}

template <typename T, typename U> vector<int> pseudoWordsBalance(const vector<U>& orig, const vector<U>& words, int nloop = - 1) {
  cerr << "pseudoWordsBalance : " << orig.size() << ", " << words.size() << flush;
  SimpleMatrix<T> result(words.size(), orig.size());
  for(int i = 0; i < orig.size(); i ++)
    result.setCol(i, countWords<T, U>(orig[i], words));
  
  vector<int> vres;
  if(nloop <= 0)
    nloop = result.cols();
  
  for(int i = 0; i < min(int(result.cols()), nloop); i ++) {
    vector<pair<T, int> > scores;
    for(int j = 0; j < result.cols(); j ++)
      scores.emplace_back(make_pair(- result.col(j).dot(result.col(j)), j));
    sort(scores.begin(), scores.end());
    if(scores[0].first == T(0))
      break;
    const int& idx(scores[0].second);
    vres.emplace_back(idx);
    for(int k = 0; k < result.rows(); k ++)
      if(result(k, idx) != T(0))
        result.row(k) *= T(0);
    T sum(0);
    for(int j = 0; j < result.rows(); j ++)
      sum += result.row(j).dot(result.row(j));
    cerr << ":" << sum << flush;
  }
  cerr << endl;
  return vres;
}

template <typename T, typename U> ostream& predTOC(ostream& os, const U& input, const vector<U>& detailtitle, const vector<U>& detail, const vector<U>& delimiter, const int& szwindow, const int& nrwords0, const T& redig = T(1)) {
  assert(detailtitle.size() == detail.size());
  const int nrwords(sqrt(T(nrwords0)));
  cerr << "predTOC: " << flush;
  vector<SimpleSparseTensor<T> > in;
  T threshin(int(0));
  vector<int> idx;
  for(int i = - int(- log(SimpleMatrix<T>().epsilon()) / log(T(int(2))) );
          i <= 0; i ++) {
    vector<corpus<T, U> > istats;
    threshin = T(int(1)) - pow(T(int(2)), - T(abs(i)));
    idx.resize(0);
    istats.resize(input.size() / (szwindow / 2));
    for(int j = 0; j < istats.size(); j ++) {
      getDetailed<T, U>(istats[j], input, j, detailtitle, detail, delimiter, szwindow, threshin);
      istats[j].reDig(redig);
      istats[j].absfy();
      auto lidx(istats[j].countIdx());
      idx.insert(idx.end(), lidx.begin(), lidx.end());
    }
    sort(idx.begin(), idx.end());
    idx.erase(unique(idx.begin(), idx.end()), idx.end());
    cerr << threshin << " : " << idx.size() << endl;
    if(nrwords <= idx.size()) {
      in.resize(istats.size());
      for(int j = 0; j < istats.size(); j ++)
        in[j] = move(istats[j].corpust);
      break;
    }
  }
  os << input;
  corpus<T, U> pstats;
  auto p(predSTen<T>(in, idx));
  pstats.corpust = move(p.first);
  getAbbreved<T>(pstats, detailtitle, detail, delimiter);
  os << pstats.simpleThresh(threshin).serialize();
  pstats.corpust = move(p.second);
  getAbbreved<T>(pstats, detailtitle, detail, delimiter);
  os << " --- or --- " << pstats.simpleThresh(threshin).serialize();
  return os;
}

// N.B. start p2/persistent.cc
template <typename T> static inline T pseudoerfscale(const T& x) {
  return sgn<T>(x) == T(int(0)) ? x : sgn<T>(x) * exp(- x * x);
}

template <typename T> static inline T pseudoierfscale(const T& y) {
  return sgn<T>(y) == T(int(0)) ? y : sgn<T>(y) * sqrt(abs(- log(abs(y))));
}

template <typename T> static inline T expscale(const T& x) {
  return sgn<T>(x) * (exp(abs(x)) - T(int(1)));
}

template <typename T> static inline T logscale(const T& x) {
  return sgn<T>(x) * log(abs(x) + T(int(1)));
}

// N.B. eliminated Persistent class and their functions.
//      we cannot avoid such a error recusrive prediction's white noise with
//      the method we once have. it's only accuracy limit dependant but reverse
//      of them doesn't work.

#define _SIMPLELIN_
#endif